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• Barycentric coordinates
  <i><b>Circles</b></i> The radical axis of two circles given by equations of this form is:
  25 KB (5,067 words) - 22:15, 31 March 2024
• Mathtime Version 1 Issue 2
  ...des. What is the ratio of the sum of the areas of all <math>4</math> small circles to the area of the large circle? (Proposed by SP343) ...the four circles. If the radius of a circle is <math>5</math> and all the circles are congruent, what is the length of the rope? (Proposed by SP343)
  15 KB (2,444 words) - 21:46, 1 January 2012
• Mock Geometry AIME 2011 Problems/Problem 8
  Two circles <math>\omega_1,\omega_2</math> have center <math>O_1,O_2</math> and radius .../math>, such that <math>CD</math> is the smallest distance connecting both circles. Then <math>O_1C+CD+DO_2 \geq O_1O_2</math>; they are both paths from <math
  3 KB (522 words) - 21:25, 3 January 2012
• Mock Geometry AIME 2011 Problems/Problem 10
  ...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>
  2 KB (324 words) - 22:28, 26 December 2022
• 1950 AHSME Problems/Problem 44
  \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}</math> ...he graph of <math>y=\log x</math>, one can clearly see that there are many circles centered at the origin that do not intersect the graph of <math>y=\log x</m
  1 KB (188 words) - 16:16, 9 May 2015
• 2012 AMC 10A Problems/Problem 11
  .../math> and <math>3</math>, respectively. A line externally tangent to both circles intersects ray <math>AB</math> at point <math>C</math>. What is <math>BC</m Let <math>D</math> and <math>E</math> be the points of tangency on circles <math>A</math> and <math>B</math> with line <math>CD</math>. <math>AB=8</ma
  2 KB (291 words) - 18:41, 22 April 2024
• 2012 AMC 10A Problems
  .../math> and <math>3</math>, respectively. A line externally tangent to both circles intersects ray <math>AB</math> at point <math>C</math>. What is <math>BC</m ...math>\frac{2\pi}{3}</math>, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side <math>2</math>. What is
  13 KB (1,994 words) - 01:31, 22 December 2023
• 2012 AMC 10A Problems/Problem 18
  ...math>\frac{2\pi}{3}</math>, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area encl Draw the hexagon between the centers of the circles, and compute its area <math>(6)(0.5)(2\sqrt{3})=6\sqrt{3}</math>. Then add
  5 KB (775 words) - 22:33, 22 October 2023
• 2012 AMC 12A Problems
  ...math>\frac{2\pi}{3}</math>, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side <math>2</math>. What is ...> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</math> and <math>Y</math>. Point <math>Z</math> in the ext
  14 KB (2,197 words) - 13:34, 12 August 2020
• 2012 AMC 12A Problems/Problem 12
  The circles have radii of <math>1</math> and <math>2</math>. Draw the triangle shown in
  3 KB (574 words) - 20:42, 3 January 2020
• 2012 AMC 12A Problems/Problem 16
  ...> has its center <math>O</math> lying on circle <math>C_2</math>. The two circles meet at <math>X</math> and <math>Y</math>. Point <math>Z</math> in the ext
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AMC 10B Problems
  ...circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
  18 KB (2,350 words) - 18:48, 9 July 2023
• 2010 AMC 8 Problems/Problem 19
  ...ne{AD}</math> has length <math>16</math>. What is the area between the two circles? ...the Pythagorean Theorem, <math>CB=6</math>. Thus the area between the two circles is
  2 KB (279 words) - 09:04, 10 March 2023
• 2012 AIME I Problems
  At each of the sixteen circles in the network below stands a student. A total of <math>3360</math> coins a Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilater
  10 KB (1,617 words) - 14:49, 2 June 2023
• 2012 AIME I Problems/Problem 7
  At each of the sixteen circles in the network below stands a student. A total of <math>3360</math> coins a
  6 KB (1,058 words) - 01:49, 25 November 2023
• 2012 AIME I Problems/Problem 13
  Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilater ...ea. We have two cases to consider; either the center <math>O</math> of the circles lies in the interior of triangle <math>ABC</math> or it does not (and we sh
  11 KB (1,889 words) - 20:42, 25 January 2023
• 2012 AIME II Problems
  ...math>\gamma</math> be the circle with diameter <math>\overline{DE}</math>. Circles <math>\omega</math> and <math>\gamma</math> meet at <math>E</math> and a se
  7 KB (1,228 words) - 12:16, 13 March 2020
• 2012 AIME II Problems/Problem 15
  ...math>\gamma</math> be the circle with diameter <math>\overline{DE}</math>. Circles <math>\omega</math> and <math>\gamma</math> meet at <math>E</math> and a se Finally, we notice there's circles! Classic setup for inversion! Since we're involving an angle-bisector, the
  13 KB (2,298 words) - 12:56, 10 September 2023
• Mock AIME I 2012 Problems
  Let two circles <math>O_1, O_2</math> with radii <math>3, 5</math> in the plane be centered
  7 KB (1,309 words) - 11:13, 8 April 2012
• Mock AIME I 2012 Problems/Problem 7
  Let two circles <math>O_1, O_2</math> with radii <math>3, 5</math> in the plane be centered ...ell, so it follows that <math>P</math> lies on the radical axis of the two circles. However, <math>X, Q, Y</math> also lie on this radical axis, so <math>P, X
  2 KB (380 words) - 17:38, 7 April 2012

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