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• Formulas relating the number of lines, sections, and intersection points
  4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections cre
  1 KB (237 words) - 17:21, 4 June 2013
• 2012 IMO Problems/Problem 5
  Since the center of <math>\Gamma</math> lies on <math>BC</math>, the three circles above are coaxial to line <math>CD</math>.
  2 KB (437 words) - 08:30, 20 November 2023
• 2005 PMWC Problems/Problem I4
  The larger circle has radius 12 cm. Each of the six identical smaller circles
  986 bytes (153 words) - 19:15, 6 October 2013
• Mock AIME 6 2006-2007 Problems
  Consider two circles of different sizes that do not intersect. The smaller circle has center <m
  7 KB (1,173 words) - 21:04, 7 December 2018
• 2014 AIME II Problems/Problem 8
  ...externally tangent and all internally tangent to a large circle. The small circles have radii <math>r</math>, <math>r</math>, and <math>3r</math>, and the big ...rcle <math>E</math> over <math>\overline{AB}</math>. Now, we have our four circles to apply that theorem. First, lets scale our image down such that Circle <m
  6 KB (826 words) - 21:31, 9 January 2024
• 1967 AHSME Problems/Problem 20
  .... 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 ...he circles. On the other hand, the areas of the squares (and areas of the circles) form a geometric sequence with common ratio <math>R^2</math>.
  2 KB (390 words) - 01:40, 16 August 2023
• Geometry Solutions
  We want to find the area of the intersection of the circles in this figure: Lets call the radius of each of the circles 1, because we are calculating probability.
  23 KB (3,182 words) - 12:30, 5 April 2014
• 2013 IMO Problems/Problem 4
  ...</math>), let its circumcircle be <math>\omega_3</math>. Then each pair of circles' radical axises, <math>BN, TW,</math> and <math>MC</math>, must concur at t
  11 KB (1,991 words) - 01:31, 19 November 2023
• 2016 AMC 10A Problems
  ...with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?
  14 KB (2,104 words) - 21:35, 7 June 2024
• 1967 IMO Problems/Problem 4
  ...,Z</math> are a permutation of <math>A,B,C</math>. Now construct the three circles <math>\mathcal C_A=(B_0PC_0),\mathcal C_B=(C_0PA_0),\mathcal C_C=(A_0PB_0)<
  2 KB (368 words) - 13:15, 29 January 2021
• 1973 IMO Problems/Problem 4
  ...be <math>E</math>. Let the height of the triangle be <math>h</math>. Draw circles around points <math>B</math> and <math>C</math> with radius <math>\frac{h}{
  3 KB (502 words) - 23:48, 12 December 2022
• 1952 AHSME Problems/Problem 8
  Two equal circles in the same plane cannot have the following number of common tangents. Two congruent coplanar circles will either be tangent to one another (resulting in <math> 3 </math> common
  1 KB (211 words) - 15:42, 2 January 2014
• 1967 AHSME Problems/Problem 33
  In this diagram semi-circles are constructed on diameters <math>\overline{AB}</math>, <math>\overline{AC
  2 KB (232 words) - 01:40, 16 August 2023
• 2014 AMC 10A Problems/Problem 12
  ...<math>\dfrac{1}{3}</math> of a circle with radius 3. There are 2 of these circles in total, so the total area of them would be <math>18\pi</math>. Now, we have to subtract the area of the circles from the total area of the hexagon, but we see that only answer (C) has <ma
  3 KB (482 words) - 11:50, 7 September 2021
• 1983 IMO Problems/Problem 2
  ...math> and <math>O_2</math> respectively. One of the common tangents to the circles touches <math>C_1</math> at <math>P_1</math> and <math>C_2</math> at <math> ...be one of the two distinct points of intersection of two unequal coplanar circles <math>C_1</math> and <math>C_2</math> with centers <math>O_1</math> and <ma
  7 KB (1,267 words) - 23:35, 29 January 2021
• 1960 AHSME Problems
  The locus of the centers of all circles of given radius <math>a</math>, in the same plane, passing through a fixed \textbf{(E) }\text{two circles} </math>
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12A Problems
  Two circles intersect at points <math>A</math> and <math>B</math>. The minor arcs <mat
  12 KB (1,863 words) - 19:04, 11 April 2024
• 2014 AMC 12A Problems/Problem 12
  Two circles intersect at points <math>A</math> and <math>B</math>. The minor arcs <mat Let the radius of the larger and smaller circles be <math>x</math> and <math>y</math>, respectively. Also, let their centers
  7 KB (1,191 words) - 23:37, 23 June 2022
• 2014 AMC 12A Problems/Problem 17
  ...a cross section and you will see that h is made up of the two radii of the circles plus some radical expression. The only choice satisfying this condition is
  4 KB (602 words) - 02:42, 13 June 2022
• 2014 AMC 10B Problems
  Two concentric circles have radii <math>1</math> and <math>2</math>. Two points on the outer circl
  13 KB (2,011 words) - 21:54, 8 November 2022

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