Create the page "Circles" on this wiki! See also the search results found.

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

View (previous 100 | next 100) (20 | 50 | 100 | 250 | 500)