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Page title matches

• Math circles
  ...around North America. Inspired by Eastern European teaching models, math circles often take a problem-discussion approach to teaching and learning [[mathema == Math Circles by City ==
  3 KB (436 words) - 18:35, 26 August 2021
• What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?
  ...ber of points of intersection that can occur when <math>2</math> different circles and <math>2</math> different straight lines are drawn on the same piece of ...rcles in <math>4</math> points. Draw another line which intersects the two circles in <math>4</math> points and also intersects the first line. There are <mat
  1 KB (188 words) - 17:11, 7 August 2020
• PaperMath’s circles
  ==PaperMath’s circles== ...of the congruent circles and where <math>n</math> is the number of tangent circles.
  4 KB (665 words) - 23:04, 27 March 2024

Page text matches

• 2005 AMC 12A Problems/Problem 16
  ...ius <math>r > s</math> is tangent to both axes and to the second and third circles. What is <math>r/s</math>?
  2 KB (307 words) - 15:30, 30 March 2024
• California mathematics competitions
  * [[Math circles]] -- There are many in California.
  4 KB (514 words) - 04:02, 21 September 2023
• Mathematics websites
  * [http://www.geometer.org/mathcircles/ Tom Davis's] site for [[math circles]] topics. * [http://www.geometer.org/mathcircles/ Tom Davis's] site for [[math circles]] topics.
  4 KB (516 words) - 03:01, 13 April 2023
• Math books
  ...//www.amazon.com/exec/obidos/ASIN/0821804308/artofproblems-20 Mathematical Circles] -- A wonderful peak into Russian math training.
  24 KB (3,177 words) - 12:53, 20 February 2024
• Math circles
  ...around North America. Inspired by Eastern European teaching models, math circles often take a problem-discussion approach to teaching and learning [[mathema == Math Circles by City ==
  3 KB (436 words) - 18:35, 26 August 2021
• San Diego Math Circle
  * [[Math circles]]
  713 bytes (94 words) - 13:36, 10 June 2008
• ARML historical results
  |[[Toronto Math Circles]] (Toronto Math Circles B1)
  19 KB (2,632 words) - 14:31, 12 June 2022
• Circle
  ==Lines in Circles== *How many circles with radius <math>r</math> can we fit around a circle with radius <math>r</
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  *Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </ma
  5 KB (892 words) - 21:52, 1 May 2021
• Planar figure
  327 bytes (41 words) - 22:27, 24 April 2008
• Homothety
  ...n use properties of similarity. Additionally, similarity (especially with circles) where parallel lines are used can indicate that homothety can be used, and
  3 KB (532 words) - 01:11, 11 January 2021
• Area
  6 KB (1,181 words) - 22:37, 22 January 2023
• 2006 AIME I Problems
  Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as s
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 10
  ...ed circles splitting into congruent areas, and there are an additional two circles on each side. The line passes through <math>\left(1,\frac 12\right)</math> Assume that if unit [[square]]s are drawn circumscribing the circles, then the line will divide the area of the [[concave]] hexagonal region of
  4 KB (731 words) - 17:59, 4 January 2022
• 2021 JMC 10
  ...ers that are <math>\tfrac{4}{3}</math> units apart. Two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> of radius <math>r_1</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  .../math> and <math> \overline{BC}</math> are common external tangents to the circles. What is the area of hexagon <math> AOBCPD</math>?
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  ...nally tangent circles, as shown. What is the sum of the areas of the three circles? ...and <math>8</math>, respectively. A common internal tangent intersects the circles at <math>C</math> and <math>D</math>, respectively. Lines <math>AB</math> a
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  ...ius <math>r > s</math> is tangent to both axes and to the second and third circles. What is <math>r/s</math>?
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  ...ly tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2002 AMC 12A Problems
  ...t circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the Let <math>C_1</math> and <math>C_2</math> be circles defined by <math>(x-10)^2 + y^2 = 36</math> and <math>(x+15)^2 + y^2 = 81</
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2002 AMC 12B Problems
  ...in a plane. What is the maximum number of points where at least two of the circles intersect?
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2004 AMC 12B Problems
  An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b</math> and <math>c</math>, with <math>b>c
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...h>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The r
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 15
  .../math> and <math> \overline{BC}</math> are common external tangents to the circles. What is the area of hexagon <math> AOBCPD</math>? ...> and <math>\angle ADP</math> are right angles due to being tangent to the circles, and the altitude creates <math>\angle OHD</math> as a right angle. <math>A
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12A Problems/Problem 13
  ...y tangent [[circle]]s, as shown. What is the sum of the areas of the three circles?
  1 KB (184 words) - 13:57, 19 January 2021
• 2006 AMC 12A Problems/Problem 16
  ...[[common internal tangent line | common internal tangent]] intersects the circles at <math>C</math> and <math>D</math>, respectively. Lines <math>AB</math> a
  2 KB (286 words) - 10:16, 19 December 2021
• 2006 AMC 12A Problems/Problem 19
  ...ctively. The equation of a common external [[tangent line|tangent]] to the circles can be written in the form <math>y=mx+b</math> with <math>m>0</math>. What
  2 KB (253 words) - 22:52, 29 December 2021
• 2005 AMC 12B Problems/Problem 5
  ...h>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> foot centered at each corner of the tile. The r
  2 KB (223 words) - 14:30, 15 December 2021
• 2005 AMC 12B Problems/Problem 16
  ...in the coordinate plane. To find the circle enclosing these <math>4</math> circles, notice that if you connect the <math>4</math> centers as a square, the dia
  2 KB (364 words) - 04:54, 16 January 2023
• 2006 AMC 10A Problems
  ...le of radius 2. The sides of <math>\triangle ABC</math> are tangent to the circles as shown, and the sides <math>\overline{AB}</math> and <math>\overline{AC} ...and <math>8</math>, respectively. A common internal tangent intersects the circles at <math>C</math> and <math>D</math>, respectively. Lines <math>AB</math> a
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 16
  ...<math>2</math>. The sides of <math>\triangle ABC</math> are tangent to the circles as shown, and the sides <math>\overline{AB}</math> and <math>\overline{AC} Let the centers of the smaller and larger circles be <math>O_1</math> and <math>O_2</math> , respectively.
  5 KB (732 words) - 23:19, 19 September 2023
• 2005 AIME II Problems
  ...math> C_2 </math> are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of <math> C_3 </math> is also a common external Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </ma
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 8
  ...> C_2 </math> are 4 and 10, respectively, and the [[center]]s of the three circles are all [[collinear]]. A [[chord]] of <math> C_3 </math> is also a common e ...e the centers and <math>r_1 = 4, r_2 = 10,r_3 = 14</math> the radii of the circles <math>C_1, C_2, C_3</math>. Let <math>T_1, T_2</math> be the points of tang
  4 KB (693 words) - 13:03, 28 December 2021
• Problem solving
  ...from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of pow
  6 KB (1,039 words) - 17:43, 30 July 2018
• Mathematician
  Contrary to the belief in some circles, "mathematician" is not synonymous with "professor of mathematics", althoug
  918 bytes (123 words) - 10:42, 30 July 2006
• 2005 AIME I Problems
  ...e area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \rfloor. </math>
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 1
  ...e area of the region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \rfloor</math> (the [[floor function]]). Define the radii of the six congruent circles as <math>r</math>. If we draw all of the radii to the points of external ta
  1 KB (213 words) - 13:17, 22 July 2017
• 2005 AIME II Problems/Problem 15
  ...th> have center <math>(x,y)</math> and radius <math>r</math>. Now, if two circles with radii <math>r_1</math> and <math>r_2</math> are externally tangent, th .... In particular, the locus of points <math>C</math> that can be centers of circles must be an ellipse with foci <math>F_1</math> and <math>F_2</math> and majo
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 4
  ...>. Thus, they enclose the area of the square minus the area of the quarter circles, which is <math>4-\pi \approx 0.86</math>, so <math>100k = \boxed{086}</mat
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME II Problems/Problem 12
  ...circle contained within the trapezoid is [[tangent]] to all four of these circles. Its radius is <math> \frac{-k+m\sqrt{n}}p, </math> where <math> k, m, n, <
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems
  ...h> A circle contained within the trapezoid is tangent to all four of these circles. Its radius is <math> \frac{-k+m\sqrt{n}}p, </math> where <math> k, m, n, <
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  In the adjoining figure, two circles with radii <math>8</math> and <math>6</math> are drawn with their centers <
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1984 AIME Problems
  ...one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this li
  6 KB (933 words) - 01:15, 19 June 2022
• 1993 AIME Problems
  ...</math>. Let <math>R\,</math> and <math>S\,</math> be the points where the circles inscribed in the triangles <math>ACH\,</math> and <math>BCH^{}_{}</math> ar
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1995 AIME Problems
  ...th>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1997 AIME Problems
  Circles of radii 5, 5, 8, and <math>m/n</math> are mutually externally tangent, whe
  7 KB (1,098 words) - 17:08, 25 June 2020
• 2002 AIME I Problems
  ...congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in th
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...circle of radius 1 is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color
  6 KB (965 words) - 16:36, 8 September 2019
• 2002 AIME II Problems
  ...he line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  Find the area of rhombus <math>ABCD</math> given that the radii of the circles circumscribed around triangles <math>ABD</math> and <math>ACD</math> are <m
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 14
  In the adjoining figure, two circles with radii <math>8</math> and <math>6</math> are drawn with their centers < ...math>QP = PR = x</math>. Extend the line containing the centers of the two circles to meet <math>R</math>, and to meet the other side of the large circle at a
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1984 AIME Problems/Problem 6
  ...side of the line is equal to the total [[area]] of the parts of the three circles to the other side of it. What is the [[absolute value]] of the [[slope]] of ...e [[midpoint]] of <math>\overline{AC}</math> (the centers of the other two circles), and call it <math>M</math>. If we draw the feet of the [[perpendicular]]s
  6 KB (1,022 words) - 19:29, 22 January 2024
• 2006 AMC 10B Problems/Problem 4
  Circles of diameter <math>1</math> inch and <math>3</math> inches have the same cen
  1 KB (172 words) - 10:47, 19 December 2021
• 1991 AIME Problems/Problem 12
  ...en we found <math>AP</math>, the segment <math>OB</math> is tangent to the circles with diameters <math>AO,CO</math>.
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...es must be tangent on the larger circle. Now consider two adjacent smaller circles. This means that the line connecting the radii is a segment of length <math ...3})^{2} = \pi (7 - 4 \sqrt {3})</math>, so the area of all <math>12</math> circles is <math>\pi (84 - 48 \sqrt {3})</math>, giving an answer of <math>84 + 48
  4 KB (740 words) - 19:33, 28 December 2022
• 1992 AIME Problems/Problem 10
  ...ique area of the two circles. We can do this by adding the area of the two circles and then subtracting out their overlap. There are two methods of finding th 2. Consider that the circles can be converted into polar coordinates, and their equations are <math>r =
  2 KB (323 words) - 12:05, 16 July 2019
• 2006 AMC 10B Problems/Problem 24
  ...at <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>?
  4 KB (558 words) - 14:38, 6 April 2024
• 1993 AIME Problems/Problem 15
  ...</math>. Let <math>R\,</math> and <math>S\,</math> be the points where the circles inscribed in the triangles <math>ACH\,</math> and <math>BCH^{}_{}</math> ar
  3 KB (449 words) - 21:39, 21 September 2023
• 1994 AIME Problems/Problem 15
  ...he respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overline{AB}, \overline{BC}, \overline{CA}</math>. ...</math> (shaded region below) is simply the sum of two [[segment]]s of the circles. If we construct the midpoints of <math>M_1, M_2 = \overline{AB}, \overline
  4 KB (717 words) - 22:20, 3 June 2021
• 1995 AIME Problems/Problem 4
  ...th>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. We label the points as following: the centers of the circles of radii <math>3,6,9</math> are <math>O_3,O_6,O_9</math> respectively, and
  3 KB (605 words) - 11:30, 5 May 2024
• 1997 AIME Problems/Problem 14
  ...</math> with radius <math>\sqrt{2+\sqrt{3}}</math>. The equations of these circles are <math>(x-1)^2 = 1</math> and <math>x^2 + y^2 = 2 + \sqrt{3}</math>. Sol
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 4
  ...f the circle in question and the segment connecting the centers of the two circles of radii <math>5</math>. By the [[Pythagorean Theorem]], we now have two eq ...give the curvature for the circle internally tangent to each of the other circles. Using Descartes' theorem, we get <math>k_4=\frac15+\frac15+\frac18+2\sqrt{
  2 KB (354 words) - 22:33, 2 February 2021
• 1998 AIME Problems/Problem 10
  3 KB (496 words) - 13:02, 5 August 2019
• 1999 AIME Problems/Problem 6
  To find the area between the circles (actually, parts of the circles), we need to figure out the [[angle]] of the [[arc]]. This could be done by
  2 KB (354 words) - 16:42, 20 July 2021
• 2000 AIME I Problems/Problem 13
  ...way in a circle with this radius centered at <math>(x,0)</math>. All these circles are [[homothety|homothetic]] with respect to a center at <math>(5,0)</math>
  3 KB (571 words) - 00:38, 13 March 2014
• 2002 AIME I Problems/Problem 2
  ...gruent [[circle]]s arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in th Let the [[radius]] of the circles be <math>r</math>. The longer dimension of the rectangle can be written as
  2 KB (287 words) - 19:54, 4 July 2013
• 2003 AIME I Problems/Problem 2
  ...dius <math>1</math> is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color To get the green area, we can color all the circles of radius <math>100</math> or below green, then color all those with radius
  4 KB (523 words) - 15:49, 8 March 2021
• 2002 AIME II Problems/Problem 15
  ...he line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b ...respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• Inradius
  ...that tangent point is equal to A to the other tangent point (explained in circles) and etc for B and C. After doing it for B and C, C (the hypotenuse) should
  2 KB (336 words) - 00:44, 23 April 2024
• Convex set
  1 KB (227 words) - 22:47, 1 March 2008
• 2006 AMC 10B Problems
  Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is ...at <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the concave hexagon <math>AOBCPD</math>?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...h> 1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math> 1/2</math> foot centered at each corner of the tile. The r
  12 KB (1,874 words) - 21:20, 23 December 2020
• Furman University Wylie Mathematics Tournament/1996 Senior Exam/Solutions
  ...is of length 2 (from equilateral triangles). Let the radius of each of the circles be <math> r. </math> Drawing <math> r </math> to the tangents of the circle
  9 KB (1,364 words) - 15:59, 21 July 2006
• University of South Carolina High School Math Contest/1993 Exam/Problem 3
  If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they ...gap in question. The area of the three gaps is half the area of one of the circles, and is thus <math>\frac{\pi}{2}</math>. The area of the whole triangle is
  1 KB (173 words) - 17:09, 4 October 2016
• University of South Carolina High School Math Contest/1993 Exam/Problem 25
  ...ngent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> a
  1 KB (172 words) - 00:11, 16 February 2016
• University of South Carolina High School Math Contest/1993 Exam/Problems
  If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they ...ngent to the circle <math>x^2 + y^2 = 4</math> at <math>(0,2)</math>? (Two circles are tangent at a point <math>P</math> if they intersect at <math>P</math> a
  14 KB (2,102 words) - 22:03, 26 October 2018
• Mock AIME 1 2006-2007 Problems/Problem 10
  ...t <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> ha
  1 KB (236 words) - 23:58, 24 April 2013
• Mock AIME 1 2006-2007/Problems
  ...t <math>A'</math>, <math>B'</math>, and <math>C'</math>, respectively. Let circles <math>A''</math>, <math>B''</math>, <math>C''</math>, and <math>I</math> ha
  8 KB (1,355 words) - 14:54, 21 August 2020
• Expected value
  ...as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of <math>n</math>.
  5 KB (789 words) - 20:56, 10 May 2024
• 1959 IMO Problems
  ...ents <math>AM </math> and <math>MB </math> as their respective bases. The circles about these squares, with respective centers <math>P </math> and <math>Q </
  3 KB (480 words) - 11:57, 17 September 2012
• 1959 IMO Problems/Problem 4
  ...e around this new point going through O and M. The intersection of the two circles is the desired third vertex of the triangle with the given hypotenuse c.
  6 KB (939 words) - 17:31, 15 July 2023
• 1959 IMO Problems/Problem 5
  ...ents <math>AM </math> and <math>MB </math> as their respective bases. The circles about these squares, with respective centers <math>P </math> and <math>Q </
  2 KB (408 words) - 01:40, 2 January 2023
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  We use the lemma that given two non-coplanar circles in space that intersect at two points, there exists a point P such that P i ...ore, <math>l_1</math> and <math>l_2</math> lie on one plane. Since our two circles are not coplanar, <math>l_1</math> and <math>l_2</math> must intersect at o
  3 KB (509 words) - 23:22, 15 August 2012
• Geometry/Introduction
  1 KB (122 words) - 16:25, 18 May 2021
• Tangent (geometry)
  ==Tangents to Circles== ...the [[radius]] that passes through the point of tangency. Any two disjoint circles have four tangents in common, two internal and two external.
  2 KB (332 words) - 21:54, 11 March 2024
• Similarity (geometry)
  * All circles are similar.
  2 KB (261 words) - 20:42, 25 November 2023
• Angle
  4 KB (597 words) - 18:39, 9 May 2024
• 1970 IMO Problems
  ...et <math>q_1, q_2</math>, and <math>q</math> be the radii of the exscribed circles of the same triangles that lie in <math>\angle ACB</math>. Prove that
  3 KB (558 words) - 00:17, 10 December 2022
• 1970 IMO Problems/Problem 1
  ...et <math>q_1, q_2</math>, and <math>q</math> be the radii of the exscribed circles of the same triangles that lie in the angle <math>ACB</math>. Prove that
  2 KB (380 words) - 22:12, 19 May 2015
• Mock AIME 5 2005-2006 Problems
  ...hin one of the smaller squares, it also lands completely within one of the circles. Let <math>P</math> be the probability that, when flipped onto the grid, th
  8 KB (1,370 words) - 21:52, 27 February 2007
• 1981 IMO Problems
  Three congruent circles have a common point <math> \displaystyle O </math> and lie inside a given t
  3 KB (379 words) - 15:09, 29 October 2006
• 1981 IMO Problems/Problem 5
  ...h>, and sides <math>a,b,c</math>, respectively, and let the centers of the circles inscribed in the [[angle]]s <math>A,B,C</math> be denoted <math>O_A, O_B, O Suppose 3 congruent circles with centres P,Q,R lie inside ABC and are such that the circle with centre
  2 KB (373 words) - 23:09, 29 January 2021
• Concentric
  190 bytes (28 words) - 20:07, 3 November 2006
• Euler line
  [[File:3 circles Euler line.png|500px|right]] ...ough <math>A, A_1</math> and is tangent to the radius AO. Similarly define circles <math>\omega_2</math> and <math>\omega_3.</math>
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2004 AMC 10A Problems
  Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is <math ...ly tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has
  15 KB (2,092 words) - 20:32, 15 April 2024
• 2005 Alabama ARML TST Problems/Problem 3
  ...circle. Then we have <math>R^2-r^2=300</math>. If the center of these two circles is <math>O</math>, the [[vertex | vertices]] are <math>A, B</math> and <mat
  1 KB (221 words) - 19:38, 6 February 2010
• 2005 Cyprus Seniors TST/Day 1/Problem 3
  (i) The circumscribed circles of the triangles <math>PP_{1}\Delta</math> and <math>PP_{2}\Gamma</math> in
  2 KB (294 words) - 15:12, 17 December 2006
• Sphere
  1 KB (247 words) - 20:36, 11 December 2020
• 2007 AIME II Problems/Problem 9
  Use the [[Two Tangent Theorem]] on <math>\triangle BEF</math>. Since both circles are inscribed in congruent triangles, they are congruent; therefore, <math>
  5 KB (818 words) - 11:05, 7 June 2022
• Mock AIME 3 Pre 2005 Problems
  Three circles are mutually externally tangent. Two of the circles have radii <math>3</math> and <math>7</math>. If the area of the triangle f Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite si
  7 KB (1,135 words) - 23:53, 24 March 2019
• Mock AIME 3 Pre 2005 Problems/Problem 1
  Three circles are mutually externally tangent. Two of the circles have radii <math>3</math> and <math>7</math>. If the area of the triangle f
  795 bytes (129 words) - 10:22, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 14
  Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite si Let <math>r_1</math> and <math>r_2</math> be the radii of circles <math>\omega_1</math> and <math>\omega_2</math> respectively.
  3 KB (563 words) - 02:05, 25 November 2023
• 2007 AIME I Problems
  ...ent to the hypotenuse and to the extension of leg <math>CB</math>, and the circles are externally tangent to each other. The length of the radius of either c
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 9
  ...to the hypotenuse and to the extension of [[leg]] <math>CB</math>, and the circles are externally tangent to each other. The length of the radius either circ ...tangents is <math>2.4x</math>. Note that if we connect the centers of the circles we have a rectangle with sidelengths 8x and 4x. So, <math>8x+2.4x+x=34</mat
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2002 IMO Shortlist Problems/G4
  Circles <math> \displaystyle S_1 </math> and <math> \displaystyle S_2 </math> inter ...stay fixed, at half the measure of arc <math> \displaystyle QP </math> on circles <math> \displaystyle S_1 </math> and <math> \displaystyle S_2 </math>, resp
  3 KB (470 words) - 07:32, 28 March 2007
• 2007 AIME II Problems/Problem 15
  ...and <math>O_C</math>. Let the center of the circle tangent to those three circles be <math>O</math>. The [[homothety]] <math>\mathcal{H}\left(I, \frac{4-r}{4 ...h>, <math>B'</math>, <math>C'</math>, and <math>O</math> be the centers of circles <math>\omega_{A}</math>, <math>\omega_{B}</math>, <math>\omega_{C}</math>,
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 USAMO Problems/Problem 2
  ...e with radius greater than <math>\frac{1}{\sqrt{2}}</math> between those 3 circles. ...ath> must decrease in radius. Hence it is sufficient to consider 3 tangent circles. By lemma 1, there is always a circle of radius greater than <math>\frac{1}
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 6
  ...and <math>Q_{A}</math> lie on <math>AO</math> since for a pair of tangent circles, the point of tangency and the two centers are [[collinear]].
  7 KB (1,274 words) - 15:11, 31 August 2017
• 2007 USA TST Problems
  Circles <math>\omega_{1}</math> and <math>\omega_{2}</math> intersect at <math>P</m
  3 KB (519 words) - 20:59, 24 May 2007
• 2004 IMO Shortlist Problems/C2
  ...ath> be positive integers. There are given <math> \displaystyle n </math> circles in the plane. Every two of them intersect at two distinct points, and all ...rcles a distinct color, which we use for both points common to the pair of circles.
  5 KB (861 words) - 10:05, 5 June 2007
• 2004 IMO Shortlist Problems/G2
  '''Lemma.''' Let <math> \displaystyle \omega_1 , \omega_2 </math> be two circles with centers <math> \displaystyle O_1, O_2 </math>, and common points <math
  10 KB (1,539 words) - 23:37, 6 June 2007
• 2007 Cyprus MO/Lyceum/Problem 21
  ...The curved parts of the band add up to a full circumference of one of the circles, so their sum is <math>20\pi\,\mathrm{ cm}</math>. The total length of the
  1 KB (175 words) - 22:21, 23 July 2020
• Pascal's Theorem
  '''Lemma.''' Let <math>\omega_1, \omega_2 </math> be two circles which intersect at <math>M, N </math>, let <math>AB </math> be a chord of <
  4 KB (712 words) - 21:57, 25 April 2020
• Cyclic
  Because two different circles intersect in at most two points, any polygon can be inscribed in at most on
  2 KB (291 words) - 05:30, 16 August 2011
• Common internal tangent line
  ...ent of both circles that intersects the segment joining the centers of two circles. ...ommon internal tangent line | common internal tangent]] [[intersect]]s the circles at <math>C</math> and <math>D</math>, respectively. [[Line]]s <math>AB</mat
  703 bytes (106 words) - 16:23, 10 March 2019
• 2006 AIME II Problems/Problem 9
  3 KB (553 words) - 10:45, 26 August 2015
• 2006 AIME II Problems
  Circles <math> \mathcal{C}_1, \mathcal{C}_2, </math> and <math> \mathcal{C}_3 </mat
  8 KB (1,350 words) - 12:00, 4 December 2022
• 2005 PMWC Problems
  The larger circle has radius 12 cm. Each of the six identical smaller circles smaller circles?
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  ...otated in the plane, they are different from one another. At most how many circles are painted with all three colors?
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2004 AMC 10A Problems/Problem 21
  Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is <math
  3 KB (476 words) - 03:50, 23 January 2023
• Excircle
  ...ega_{A}</math> denote the circle with <math>AI_{A}</math> as its diameter. Circles <math>\omega</math> and <math>\omega_{A}</math> meet at <math>P</math> othe
  5 KB (843 words) - 03:02, 1 July 2020
• 1997 USAMO Problems/Problem 2
  ...e that the problem describes! We do this similarly for the others pairs of circles.
  2 KB (347 words) - 08:13, 26 August 2021
• 1962 IMO Problems
  ...e. Prove that the distance <math>d</math> between the centers of these two circles is
  3 KB (431 words) - 21:17, 20 August 2020
• De Longchamps point
  The de Longchamps point of a triangle is the radical center of the power circles of the triangle. Prove that De Longchamps point lies on Euler line. ...h> point <math>A'</math> with radius <math>R_A = AA'.</math> The other two circles are defined symmetrically.
  10 KB (1,780 words) - 09:23, 17 November 2022
• 2007 AMC 10A Problems
  Four circles of radius <math>1</math> are each tangent to two sides of a square and exte ...2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1959 AHSME Problems
  On the same side of a straight line three circles are drawn as follows: a circle with a radius of <math>4</math> inches is ta ...tangent to the line and to the other two circles. The radius of the equal circles is:
  22 KB (3,345 words) - 20:12, 15 February 2023
• 1966 AHSME Problems
  ...</math> feet, a possible value for the distance between the centers of the circles, expressed in feet, is:
  19 KB (3,159 words) - 22:10, 11 March 2024
• 1991 USAMO Problems/Problem 5
  ...ne <math>AB</math> at points <math>D_a,D_b</math>, respectively; let these circles touch <math>CD</math> at <math>C_a</math>, <math>C_b</math>, respectively;
  5 KB (953 words) - 12:18, 4 September 2018
• 2002 AMC 12B Problems/Problem 14
  ...[[plane]]. What is the maximum number of points where at least two of the circles intersect?<!-- don't remove the following tag, for PoTW on the Wiki front p ...ath>2</math> times. Since there are <math>{4\choose 2} = 6</math> pairs of circles, the maximum number of possible intersections is <math>6 \cdot 2 = 12</math
  2 KB (282 words) - 14:04, 12 July 2021
• 2007 AMC 10A Problems/Problem 15
  Four circles of radius <math>1</math> are each tangent to two sides of a square and exte ...eter of the big circle of radius <math>2</math> and two radii of the small circles of radius <math>1</math>. Therefore, the side length of this square is <cma
  2 KB (326 words) - 10:29, 4 June 2021
• Construction
  ...ith straight edge and compass (i.e. the ability to mark off segments, draw circles and arcs, and draw straight lines) are a branch of [[geometry]] that rely o A '''compass''' is a tool that can draw circles and arcs of circles.
  3 KB (443 words) - 20:52, 28 August 2014
• 2007 AMC 8 Problems
  Amanda draws five circles with radii <math>1, 2, 3,
  12 KB (1,800 words) - 20:01, 8 May 2023
• 2007 AMC 10A Problems/Problem 24
  ...2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</
  3 KB (439 words) - 15:39, 3 June 2021
• 1990 USAMO Problems/Problem 5
  Let <math>A'</math> be the intersection of the two circles (other than <math>A</math>). <math>AA'</math> is perpendicular to both <mat ...AA'B</math> are both <math>90^{\circ}</math>, <math>A'</math> lies on the circles with diameters <math>AC</math> and <math>AB</math>.
  3 KB (604 words) - 20:52, 24 October 2018
• Isogonal conjugate
  ==Circles== Point <math>M</math> lies on circles <math>APD</math> and <math>AEP' \implies </math> spiral similarity centered
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 12A Problems
  ...th>, and <math>\overline{CD}</math> bisects the right angle. The inscribed circles of <math>\triangle ADC</math> and <math>\triangle BCD</math> have radii <ma
  13 KB (2,025 words) - 13:56, 2 February 2021
• 2008 AMC 12A Problems/Problem 13
  Ratio of areas of circles is ratio of radii squared, so the answer is <math>\left(\frac{1}{3}\right)^ ...and <math>D</math>, respectively. Since line <math>CD</math> is tangent to circles <math>O</math> and <math>P</math>, it must be perpendicular to <math>ON</ma
  4 KB (630 words) - 20:32, 4 June 2021
• Elliptical geometry
  ...closest analogue to lines on a sphere is a great circle, and any two great circles on a sphere must intersect.
  685 bytes (113 words) - 12:22, 20 November 2012
• 2008 AMC 12A Problems/Problem 20
  ...th>, and <math>\overline{CD}</math> bisects the right angle. The inscribed circles of <math>\triangle ADC</math> and <math>\triangle BCD</math> have radii <ma
  6 KB (951 words) - 16:31, 2 August 2019
• 2007 AMC 10B Problems/Problem 18
  A circle of radius <math>1</math> is surrounded by <math>4</math> circles of radius <math>r</math> as shown. What is <math>r</math>? ...circles to get <math>r+1.</math> You can also add the radius of two outer circles and use a <math>45-45-90</math> triangle to get <math>\frac{2r}{\sqrt{2}} =
  2 KB (385 words) - 14:17, 4 June 2021
• 2004 AMC 12A Problems/Problem 19
  ...ly tangent to each other, and internally tangent to circle <math>D</math>. Circles <math>B</math> and <math>C</math> are congruent. Circle <math>A</math> has The four circles have curvatures <math>-\frac{1}{2}, 1, \frac{1}{r}</math>, and <math>\frac{
  5 KB (785 words) - 00:29, 31 July 2023
• 2004 AMC 12A Problems/Problem 24
  As the red circles move about segment <math>AB</math>, they cover the area we are looking for.
  3 KB (434 words) - 22:25, 22 November 2021
• 2008 AMC 12B Problems
  ...ing <math>(0,1)</math> to <math>(2,1)</math>. What is the probability that circles <math>A</math> and <math>B</math> intersect?
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AMC 12B Problems/Problem 21
  ...ing <math>(0,1)</math> to <math>(2,1)</math>. What is the probability that circles <math>A</math> and <math>B</math> intersect? ...sqrt{3}</math> horizontally. Thus, if <math>|A_x-B_x|<\sqrt{3}</math>, the circles intersect.
  6 KB (1,008 words) - 11:46, 24 December 2020
• 1985 AJHSME Problems
  ...f the six whole numbers <math>10\text{--}15</math> is placed in one of the circles so that the sum, <math>S</math>, of the three numbers on each side of the t
  12 KB (1,670 words) - 17:42, 24 November 2021
• 2008 AIME II Problems/Problem 13
  ...ing <math>S_2</math>, we find that <math>S</math> is the union of six unit circles centered at <math>\text{cis}\, \frac{k\pi}{6}</math>, <math>k = 0,1,2,3,4,5
  6 KB (894 words) - 18:56, 25 December 2022
• 2008 AMC 10A Problems/Problem 17
  ...<math>3 \times 6</math>, and three <math>120^{\circ}</math> degree arcs of circles of [[radius]] <math>3</math>. Thus the answer is <cmath>3(3 \times 6) + 3 \
  2 KB (348 words) - 19:59, 4 June 2021
• 2008 USAMO Problems/Problem 2
  We know that these two circles already intersect at <math>O</math> so we can reflect over the line between ...ges under the inversion of lines <math>BDF</math> and <math>CFE</math> are circles that intersect in <math>A</math> and <math>F'</math>, it follows that <math
  20 KB (3,565 words) - 11:54, 1 May 2024
• 2003 AMC 12A Problems/Problem 17
  ...0</math> is obviously referring to the x-coordinate of the point where the circles intersect at the origin, <math>D</math>, so the second value must be referr
  6 KB (1,026 words) - 22:35, 29 March 2023
• 1966 AHSME Problems/Problem 8
  ...</math> feet, a possible value for the distance between the centers of the circles, expressed in feet, is: The distance between the centers of the circles (points <math>P</math> and <math>O</math>) is the sum of the heights of <ma
  2 KB (256 words) - 01:45, 26 June 2016
• 1998 AHSME Problems
  Let <math>R</math> be a rectangle. How many circles in the plane of <math>R</math> have a diameter both of whose endpoints are
  15 KB (2,222 words) - 10:40, 11 August 2020
• 1988 IMO Problems/Problem 1
  Consider 2 concentric circles with radii <math>R</math> and <math>r</math> (<math>R>r</math>) with center
  3 KB (545 words) - 11:32, 30 January 2021
• 1984 IMO Problems/Problem 4
  ...MT = \frac{1}{2} AM \cdot UN</math> because of the equivalence of radii in circles. Hence <math>[ANM] = [DMN]</math>, so <math>A</math> and <math>D</math> are
  4 KB (750 words) - 23:49, 29 January 2021
• 1989 IMO Problems/Problem 4
  ...ion of circle <math>B</math> with the line <math>AB</math>. Thus, the two circles are externally tangent to each other. ...reasoning that we found circles <math>A,B</math> are tangent, we find that circles <math>P,A</math> and <math>P,B</math> are externally tangent also.
  4 KB (771 words) - 11:57, 30 January 2021
• 1995 IMO Problems/Problem 1
  ...<math>A,B,C,D</math> be four distinct points on a line, in that order. The circles with diameters <math>AC</math> and <math>BD</math> intersect at <math>X</ma ...h>BD</math> is line <math>DN</math>. Since the pairwise radical axes of 3 circles are concurrent, we have <math>AM,DN,XY</math> are concurrent as desired.
  5 KB (847 words) - 19:03, 12 October 2021
• 2000 USAMO Problems
  ...passing through <math>A_1</math> and <math>A_2.</math> Suppose there exist circles <math>\omega_2, \omega_3, \dots, \omega_7</math> such that for <math>k = 2,
  3 KB (495 words) - 19:02, 18 April 2014
• 2000 USAMO Problems/Problem 5
  ...passing through <math>A_1</math> and <math>A_2.</math> Suppose there exist circles <math>\omega_2, \omega_3, \dots, \omega_7</math> such that for <math>k = 2, ...heta_{1} = \theta_{7}</math> implies that <math>O_1 \equiv O_7</math>, and circles <math>\omega_1</math> and <math>\omega_7</math> are the same circle since t
  3 KB (609 words) - 09:52, 20 July 2016
• 2008 iTest Problems
  ...in a plane. What is the maximum number of points where at least two of the circles intersect?'' Tony realizes that he can draw the four circles such that each pair of circles intersects in two points. After careful doodling, Tony finds the correct an
  71 KB (11,749 words) - 01:31, 2 November 2023
• Spiral similarity
  ===Common point for 6 circles=== ...math> so these circles contain point <math>O</math>. Similarly for another circles.
  28 KB (4,863 words) - 00:29, 16 December 2023
• 2002 AMC 10A Problems
  ...t circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the
  11 KB (1,733 words) - 11:04, 12 October 2021
• 2002 AMC 10B Problems
  Circles of radius <math>2</math> and <math>3</math> are externally tangent and are ...in a plane. What is the maximum number of points where at least two of the circles intersect?
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2002 AMC 12A Problems/Problem 7
  Let <math>r_1</math> and <math>r_2</math> be the radii of circles <math>A</math> and<math> B</math>, respectively. ...rough and simplify to get <math>\frac{r_1}{r_2}=\frac{2}{3}</math>. As all circles are similar to one another, the ratio of the areas is just the square of th
  3 KB (492 words) - 14:46, 31 January 2024
• 2004 AMC 10B Problems
  An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b</math> and <math>c</math>, with <math>b>c Three circles of radius <math>1</math> are externally tangent to each other and internall
  13 KB (1,988 words) - 23:06, 7 March 2024
• 1975 USAMO Problems/Problem 4
  ...<math>AB</math> passing through <math>P</math> and terminating on the two circles such that <math>AP\cdot PB</math> is a maximum. Let <math>E</math> and <math>F</math> be the centers of the small and big circles, respectively, and <math>r</math> and <math>R</math> be their respective ra
  2 KB (410 words) - 15:13, 13 August 2014
• 1962 IMO Problems/Problem 6
  ...e. Prove that the distance <math>d</math> between the centers of these two circles is
  2 KB (308 words) - 06:29, 16 December 2023
• 1962 IMO Problems/Problem 7
  In order for these two circles to be part of the same sphere and also tangent to line <math>AB</math>, the The only way these to circles can share the same tangent point on edge <math>AB</math> is if <math>|AP_{A
  11 KB (1,928 words) - 20:52, 21 November 2023
• 1963 IMO Problems/Problem 2
  ...<math>X</math>, the total locus is the union of the circumferences of all circles that have a diameter <math>AX</math>, where <math>X</math> is some point on
  2 KB (306 words) - 04:49, 19 February 2019
• 1975 USAMO Problems
  ...<math>AB</math> passing through <math>P</math> and terminating on the two circles such that <math>AP\cdot PB</math> is a maximum.
  2 KB (332 words) - 18:57, 3 July 2013
• 2021 GMC 10B
  ...grid has 16 circles with radius of <math>\frac{1}{2}</math> such that all circles have vertices of the square as center. Assume that the diagram continues on
  11 KB (1,695 words) - 14:33, 7 March 2022
• 1998 AHSME Problems/Problem 23
  Both sets of points are quite obviously circles. To show this, we can rewrite each of them in the form <math>(x-x_0)^2 + (y ...distance between the two centers is <math>5</math>, and therefore the two circles intersect if <math>2\leq r \leq 12</math>.
  1 KB (193 words) - 09:12, 2 December 2018
• 1998 AHSME Problems/Problem 11
  Let <math>R</math> be a rectangle. How many circles in the plane of <math>R</math> have a diameter both of whose endpoints are
  929 bytes (132 words) - 14:29, 5 July 2013
• 1985 AJHSME Problems/Problem 24
  ...number|whole numbers]] <math>10-15</math> is placed in one of the [[circle|circles]] so that the sum, <math>S</math>, of the three numbers on each side of the
  3 KB (430 words) - 09:06, 22 January 2023
• 1986 AJHSME Problems
  ...the ratio of the area of the shaded region to the area of one of the small circles?
  14 KB (2,054 words) - 15:41, 8 August 2020
• 1986 AJHSME Problems/Problem 23
  ...the ratio of the area of the shaded region to the area of one of the small circles?
  2 KB (277 words) - 21:32, 3 July 2013
• 2004 AMC 10B Problems/Problem 16
  Three circles of radius <math>1</math> are externally tangent to each other and internall ...1}{r}</math> (b/c the bigger circle is externally tangent to all the other circles, the radius of the bigger circle is negative). Then, we can solve:
  2 KB (287 words) - 14:05, 5 January 2022
• 2004 AMC 10B Problems/Problem 22
  ...a circumscribed circle. What is the distance between the centers of those circles?
  5 KB (700 words) - 13:46, 6 April 2024
• 2004 AMC 10B Problems/Problem 25
  ...ture, that is outside the smaller circle and inside each of the two larger circles? Let <math>C</math> and <math>D</math> be the intersections of the two large circles. Connect them to <math>A</math> and <math>B</math> to get the picture below
  4 KB (703 words) - 22:37, 2 November 2022
• 2004 AMC 12B Problems/Problem 10
  An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b</math> and <math>c</math>, with <math>b>c<
  2 KB (340 words) - 14:35, 23 April 2023
• 2009 AMC 12A Problems
  ...around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the
  13 KB (2,105 words) - 13:13, 12 August 2020
• 2009 AMC 10A Problems
  ...s it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning and end of circle <math>B ...les is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
  14 KB (2,130 words) - 11:32, 7 November 2021
• 2009 AMC 12A Problems/Problem 19
  ...around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the ...itude we dropped to the side of each polygon) are the radii of the smaller circles.
  4 KB (630 words) - 21:27, 30 December 2023
• 2009 AMC 10A Problems/Problem 18
  ...e total number of students be <math>100</math>. Draw a venn diagram with 2 circles encompassing these 4 regions:
  3 KB (402 words) - 10:29, 2 August 2021
• 2009 AMC 10A Problems/Problem 21
  ...les is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle? Draw some of the radii of the small circles as in the picture below.
  3 KB (474 words) - 12:50, 29 September 2023
• 2009 AMC 10A Problems/Problem 19
  ...s it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning and end of circle <math>B
  2 KB (276 words) - 09:57, 8 June 2021
• 2001 AMC 12 Problems/Problem 18
  ...>\overline{AC}</math>. Moreover, the small circle is tangent to both other circles, hence we have <math>SA=1+r</math> and <math>SB=4+r</math>. ...e: This case corresponds to the other circle that is tangent to both given circles and the common tangent line. By coincidence, due to the <math>4:1</math> ra
  5 KB (822 words) - 01:35, 7 February 2024
• 2002 AMC 12A Problems/Problem 5
  ...t circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the ...s radius <math>1+1+1=3</math>, and thus area <math>9\pi</math>. The little circles have area <math>\pi</math> each; since there are 7, their total area is <ma
  1 KB (192 words) - 12:35, 8 November 2021
• 2009 AMC 10A Problems/Problem 6
  ...hat is inscribed in it. Double means that <math>2</math> of the small full circles will be able to fit the larger semi-circle. So, therefore, the area that is
  2 KB (380 words) - 09:21, 8 June 2021
• 2002 AMC 12A Problems/Problem 18
  Let <math>C_1</math> and <math>C_2</math> be circles defined by <math>(x-10)^2 + y^2 = 36</math> and <math>(x+15)^2 + y^2 = 81</ Line PQ is tangent to both circles, so it forms a right angle with the radii (6 and 9). This, as well as the t
  3 KB (485 words) - 03:13, 1 September 2023
• 2009 AIME II Problems
  ...gent to circle <math>A</math> at the other two vertices of <math>T</math>. Circles <math>B</math>, <math>C</math>, and <math>D</math> are all externally tange
  8 KB (1,366 words) - 21:33, 3 January 2021
• 2000 AMC 8 Problems
  ...und the region shown. Arcs <math>AB</math> and <math>AD</math> are quarter-circles, and arc <math>BCD</math> is a semicircle. What is the area, in square unit
  15 KB (2,165 words) - 03:32, 13 April 2024
• 2009 AIME II Problems/Problem 5
  ...gent to circle <math>A</math> at the other two vertices of <math>T</math>. Circles <math>B</math>, <math>C</math>, and <math>D</math> are all externally tange ...B</math> and <math>E</math>, and <math>Y</math> be the intersection of the circles with centers <math>C</math> and <math>E</math>. Since the radius of <math>B
  2 KB (372 words) - 20:54, 9 August 2020
• 2009 USAMO Problems/Problem 1
  Given circles <math>\omega_1</math> and <math>\omega_2</math> intersecting at points <mat
  2 KB (290 words) - 13:16, 17 April 2021
• Radical axis
  ...ical axis is the line that one gets when you subtract the equations of two circles). a. The radical axis is a line perpendicular to the line connecting the circles' centers (line <math>l</math>).
  10 KB (1,797 words) - 02:05, 24 October 2023
• 1964 IMO Problems
  ...es, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of <math>a,b,c</math>).
  3 KB (414 words) - 12:38, 29 January 2021
• 1964 IMO Problems/Problem 3
  ...es, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of <math>a,b,c</math>).
  2 KB (414 words) - 12:48, 29 January 2021
• 1967 IMO Problems
  ...alpha </math>. If <math>\triangle ABD</math> is acute, prove that the four circles of radius <math>1</math> with centers <math>A</math>, <math>B</math>, <math
  3 KB (522 words) - 12:40, 29 January 2021
• Asymptote: Drawing
  ==Circles== ...everal useful pre defined paths for drawing things like unit squares, unit circles, etc. Just use the unit- paths!
  3 KB (433 words) - 22:13, 26 April 2024
• 2009 USAMO Problems
  Given circles <math>\omega_1</math> and <math>\omega_2</math> intersecting at points <mat
  4 KB (718 words) - 18:16, 17 September 2012
• Asymptote: Graphing
  Remember, you can still draw normal functions, so you can create lines, circles and ellipses.
  2 KB (299 words) - 17:09, 15 March 2016
• 1992 AJHSME Problems
  Four circles of radius <math>3</math> are arranged as shown. Their centers are the vert
  17 KB (2,346 words) - 13:36, 19 February 2020
• Constructible number
  *(E) Finding the intersection point(s) of two previously constructed circles. *(iii) <math>\theta</math> is the intersection of circles with centers <math>z</math> and <math>z'</math> and radii <math>r</math> an
  8 KB (1,305 words) - 08:39, 21 August 2009
• 2010 AMC 10B Problems
  Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tangent to <m
  12 KB (1,817 words) - 22:44, 22 December 2020
• Mock AIME 1 2010 Problems
  Let <math>\omega_1</math> and <math>\omega_2</math> be circles of radii 5 and 7, respectively, and suppose that the distance between their
  7 KB (1,297 words) - 01:29, 25 November 2016
• 2010 AIME I Problems/Problem 15
  14 KB (2,210 words) - 13:14, 11 January 2024
• 2010 AIME II Problems/Problem 15
  ...}{54}=\frac{b^2z}{58}</math>. This equation determines the radical axis of circles <math>ANM</math> and <math>AED</math>, on which points <math>P</math> and <
  9 KB (1,523 words) - 15:24, 21 November 2023
• 1995 AJHSME Problems
  Three congruent circles with centers <math>P</math>, <math>Q</math>, and <math>R</math> are tangent
  14 KB (2,096 words) - 18:29, 2 January 2023
• 2007 AMC 8 Problems/Problem 25
  ...math>. The larger sections trisect a "ring" which is the difference of two circles, one with radius <math>3</math>, the other radius <math>6</math>. So, the a
  3 KB (516 words) - 13:40, 23 January 2024
• 2010 AMC 12B Problems/Problem 18
  ...either circle's center (symmetry, you chose!). The intersection of the two circles should form a geometrical lens shape. By sectors, ...}{2}x^2</math> and <math>x^2</math> are tangent. The area ratio of the two circles is <cmath>\frac{\pi}{4\pi} = \boxed{\frac{1}{4} \text{(C)}}</cmath>.
  6 KB (1,105 words) - 13:39, 9 January 2024
• 1969 IMO Problems
  ...he perpendicular from <math>C</math> to <math>AB</math>. We consider three circles, <math>\gamma_1, \gamma_2, \gamma_3</math>, all tangent to the line <math>A
  3 KB (428 words) - 13:34, 29 January 2021
• 2006 AMC 8 Problems
  ...th>. Circle <math> Z </math> has an area of <math> 9 \pi </math>. List the circles in order from smallest to largest radius.
  16 KB (2,215 words) - 19:18, 10 April 2024
• 2010 AMC 10B Problems/Problem 20
  Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tangent to <m
  3 KB (483 words) - 18:41, 4 May 2024
• 2011 AMC 12A Problems
  Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math Circles with radii <math>1</math>, <math>2</math>, and <math>3</math> are mutually
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 11
  Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math ...requested area is the area of <math>C</math> minus the area shared between circles <math>A</math>, <math>B</math> and <math>C</math>.
  3 KB (515 words) - 19:56, 10 August 2023
• 2011 AMC 12A Problems/Problem 17
  Circles with radii <math>1</math>, <math>2</math>, and <math>3</math> are mutually The centers of these circles form a 3-4-5 triangle, which has an area equal to 6.
  3 KB (462 words) - 17:49, 3 February 2024
• 2011 AMC 10A Problems
  Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math
  13 KB (1,903 words) - 18:09, 19 April 2021
• 1993 AHSME Problems
  ...of radius 1 that are in the same plane and tangent to each other. How many circles of radius 3 are in this plane and tangent to both <math>C_1</math> and <mat
  20 KB (2,814 words) - 08:15, 27 June 2021
• 2021 AIME II Problems/Problem 5
  ==Solution 5 (Circles)==
  14 KB (2,269 words) - 00:43, 2 January 2023
• 2007 AMC 8 Problems/Problem 16
  Amanda Reckonwith draws five circles with radii <math>1, 2, 3,
  2 KB (332 words) - 12:22, 16 August 2021
• 2002 AMC 10B Problems/Problem 5
  Circles of radius <math>2</math> and <math>3</math> are externally tangent and are A line going through the centers of the two smaller circles also goes through the diameter. The length of this line within the circle i
  2 KB (247 words) - 17:30, 5 January 2021
• Descartes' Circle Formula
  Descartes' Circle Formula is a relation held between four mutually tangent circles. ...tangent to circle B of radius <math>r_b</math>. Then the curvatures of the circles are simply the reciprocals of their radii, <math>\frac{1}{r_a}</math> and <
  2 KB (288 words) - 21:00, 24 December 2017
• 1998 USAMO Problems
  Let <math>{\cal C}_1</math> and <math>{\cal C}_2</math> be concentric circles, with <math>{\cal C}_2</math> in the interior of <math>{\cal C}_1</math>.
  3 KB (486 words) - 06:11, 24 November 2020
• Mock Geometry AIME 2011 Problems
  Two circles <math>\omega_1,\omega_2</math> have center <math>O_1,O_2</math> and radius ...h>7x+y=28.</math> Suppose that one of the tangent lines from the origin to circles <math>\omega_1</math> and <math>\omega_2</math> meets <math>\omega_1</math>
  8 KB (1,349 words) - 19:10, 14 June 2022
• 2007 AMC 10B Problems
  ...).</math> What is the area of the intersection of the interiors of the two circles? A circle of radius <math>1</math> is surrounded by <math>4</math> circles of radius <math>r</math> as shown. What is <math>r</math>?
  15 KB (2,297 words) - 12:57, 19 February 2020
• 2011 AMC 10B Problems/Problem 12
  ...s of the smaller circles, and <math>b+6</math> be the radius of the larger circles. The length of the inner edge will be <math>2a+2b \pi </math> and the lengt ...will take Keiko the same time to walk that length for both inner and outer circles. Instead, focus on the circular part. If the diameter of the smaller circle
  2 KB (337 words) - 15:15, 24 August 2021
• Olympiad books
  ...- '''San Gaku''' - '''Problem''' book (it contains lots of theorems about circles).
  17 KB (2,261 words) - 00:30, 22 April 2024
• 2007 AMC 10B Problems/Problem 13
  ...).</math> What is the area of the intersection of the interiors of the two circles?
  898 bytes (142 words) - 20:42, 15 February 2024
• 2015 IMO Problems/Problem 4
  ...circles' common chord, then the other two chords cut by these lines on the circles are parallel.''<br /> ...lemma is always <math>\Omega</math>, while the other is one of three other circles from the problem statement. Applying the lemma to the lines <math>FDH</math
  3 KB (502 words) - 23:58, 5 October 2015
• 1967 AHSME Problems
  .... If <math>S_n</math> is the sum of the areas of the first <math>n</math> circles so inscribed, then, as <math>n</math> grows beyond all bounds, <math>S_n</m In this diagram semi-circles are constructed on diameters <math>\overline{AB}</math>, <math>\overline{AC
  20 KB (3,108 words) - 14:14, 20 February 2020
• 2003 AMC 8 Problems
  The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
  16 KB (2,236 words) - 12:02, 19 February 2024
• Magic squares
  ''The Zen of Magic Squares, Circles, and Stars'' by Clifford A. Pickover
  666 bytes (109 words) - 22:53, 30 June 2021
• 2008 IMO Problems/Problem 1
  ...g their centers. We know that <math>H</math> is on the radical axis of the circles centered at <math>M_B</math> and <math>M_C</math>, so <math>A</math> is too
  2 KB (415 words) - 01:08, 19 November 2023
• 2011 IMO Problems/Problem 6
  Regardless, the circumcenter and an intersection of the circles are collinear with (0,0), so it is a tangency.
  2 KB (317 words) - 01:22, 19 November 2023
• 2000 AMC 8 Problems/Problem 19
  ...und the region shown. Arcs <math>AB</math> and <math>AD</math> are quarter-circles, and arc <math>BCD</math> is a semicircle. What is the area, in square unit ...here <math>O</math> is the center of the semicircle. You have two quarter circles on top, and two quarter circle-sized "bites" on the bottom. Move the piece
  2 KB (383 words) - 16:58, 12 January 2024
• 1996 AJHSME Problems/Problem 25
  ...er circle meaning the set of points are equidistant from the center of the circles to the outside of the big circle.
  2 KB (270 words) - 20:03, 26 May 2021
• 1995 AJHSME Problems/Problem 9
  Three congruent circles with centers <math>P</math>, <math>Q</math>, and <math>R</math> are tangent If circle <math>Q</math> has diameter <math>4</math>, then so do congruent circles <math>P</math> and <math>R</math>. Draw a diameter through <math>P</math>
  2 KB (409 words) - 00:15, 5 July 2013
• 2008 AMC 8 Problems/Problem 18
  Two circles that share the same center have radii <math>10</math> meters and <math>20</
  2 KB (210 words) - 13:37, 19 October 2020
• Inscribe
  ...ircle. Any polygon can be inscribed in at most one circle (because any two circles can intersect at no more than two points), and any polygon can have at most
  732 bytes (123 words) - 05:35, 16 August 2011
• 1979 USAMO Problems/Problem 4
  ...and radius <math>\overline{PO}.</math> Lines <math>OA,OB</math> go to the circles <math>(O_1),(O_2)</math> passing through <math>P,O</math> and the line <mat
  3 KB (568 words) - 12:24, 11 March 2018
• 1996 AHSME Problems
  ...th>1</math> has center at <math>(5,0)</math>. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the <
  15 KB (2,343 words) - 13:39, 19 February 2020
• 1996 AHSME Problems/Problem 18
  ...th>1</math> has center at <math>(5,0)</math>. A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the < The two circles are tangent to each other at the point <math>(4,0)</math>, since it is both
  2 KB (415 words) - 14:43, 5 June 2016
• 1996 AHSME Problems/Problem 27
  The spheres now become circles with centers at <math>(1,0)</math> and <math>(\frac{21}{2},0)</math>. They ...d <math>B</math> and third point at the first quadrant intersection of the circles. Let's call that altitude <math>h</math>.
  7 KB (1,046 words) - 11:42, 30 September 2023
• 1992 AJHSME Problems/Problem 24
  Four circles of radius <math>3</math> are arranged as shown. Their centers are the vert ...2=36</math>. The unshaded region of the square is composed of four quarter circles, and has the area of one circle, which is <math>9\pi</math>. The area of th
  1 KB (191 words) - 00:10, 5 July 2013
• 2010 PUMaC Problems
  Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded
  22 KB (3,694 words) - 23:58, 3 June 2022
• 1950 AHSME Problems
  ...textbf{(D)}\ \text{Cuts neither axis}\qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin} </math>
  22 KB (3,306 words) - 19:50, 3 May 2023
• 2006 AMC 8 Problems/Problem 7
  ...th>. Circle <math> Z </math> has an area of <math> 9 \pi </math>. List the circles in order from smallest to the largest radius. Using the formulas of circles, <math> C=2 \pi r </math> and <math> A= \pi r^2 </math>, we find that circl
  868 bytes (153 words) - 19:35, 14 January 2024
• 1985 AHSME Problems
  ...\triangle ABC</math> with legs <math>5</math> and <math>12</math>, arcs of circles are drawn, one with center <math>A</math> and radius <math>12</math>, the o
  17 KB (2,488 words) - 03:26, 20 March 2024
• 1985 AHSME Problems/Problem 3
  ...\triangle ABC</math> with legs <math>5</math> and <math>12</math>, arcs of circles are drawn, one with center <math>A</math> and radius <math>12</math>, the o ...> and <math>BN = BC = 5</math> since they are both radii of the respective circles. Thus <math>MB = AB-AM = 13-12 = 1</math>, and so <math>MN = BN-BM = 5-1 =
  1 KB (207 words) - 18:40, 19 March 2024
• 2010 AMC 8 Problems
  ...of a <math>12</math>-inch pizza when placed. If a total of <math>24</math> circles of pepperoni are placed on this pizza without overlap, what fraction of the ...ne{AD}</math> has length <math>16</math>. What is the area between the two circles?
  18 KB (2,768 words) - 21:05, 9 January 2024
• 2010 AMC 8 Problems/Problem 10
  ...of a <math>12</math>-inch pizza when placed. If a total of <math>24</math> circles of pepperoni are placed on this pizza without overlap, what fraction of the ...c{12}{6} = 2</math>. From that we see that the area of the <math>24</math> circles of pepperoni is <math>\left ( \frac{2}{2} \right )^2 (24\pi) = 24\pi</math>
  955 bytes (141 words) - 23:55, 22 January 2024

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