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• 2006 Canadian MO Problems/Problem 2
  Now, we use [[vector]] geometry: intersection <math>I</math> of the diagonals of <math>DEFG</math> is also ..., and <math>MY // AH</math>, so <math>MYX</math> and <math>A'AX</math> are similar, and so <math>X</math> lies on <math>A'M</math>, as desired. Reversing the
  2 KB (416 words) - 20:00, 21 September 2014
• 2002 USAMO Problems/Problem 2
  ...and [[inradius]], respectively. Prove that triangle <math>ABC </math> is similar to a triangle <math>T </math> whose side lengths are all positive integers [[Category:Olympiad Geometry Problems]]
  2 KB (298 words) - 22:32, 6 April 2016
• 2007 AIME I Problems/Problem 9
  Use a similar solution to the aforementioned solution. Instead, call <math>\angle CAB = 2 ...s that triangles <math>I_{C}AB</math> and <math>I_{C}O_{1}O_{2}</math> are similar.
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2002 IMO Shortlist Problems/G4
  ...hen we must also have all triangles <math> \displaystyle QA_1X </math> are similar. Since <math> \displaystyle Q </math> is fixed, this means that there exis [[Category:Olympiad Geometry Problems]]
  3 KB (470 words) - 07:32, 28 March 2007
• 2007 AIME II Problems/Problem 15
  Now, triangles <math>ABC</math> and <math>A'B'C'</math> are similar by parallel sides, so we can find ratios of two quantities in each triangle [[Category:Intermediate Geometry Problems]]
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2005 IMO Shortlist Problems/G1
  ...les <math> \displaystyle MCP </math>, <math> \displaystyle DBI </math> are similar. ...th> are similar. Thus triangles <math> \displaystyle MCP, NPI </math> are similar. But we note that by measures of intercepted arcs, <math> \angle ICP = \fr
  7 KB (1,088 words) - 16:57, 30 May 2007
• 2004 IMO Shortlist Problems/G2
  ...suffices to show that triangles <math> \displaystyle DFC, DAF </math> are similar. Since these triangles share a common angle, it then suffices to show <mat ...AOE </math> to <math> \displaystyle FDE </math>, i.e., these triangles are similar. Since <math> \displaystyle OA = OE </math>, it follows that <math> \displ
  10 KB (1,539 words) - 23:37, 6 June 2007
• Circumradius
  ...gle and <math>s = (a+b+c)/2</math> is the semiperimeter. Note that this is similar to the previously mentioned formula; the reason being that <math>A = rs</ma [[Category:Geometry]]
  4 KB (729 words) - 16:52, 19 February 2024
• 2007 AMC 12A Problems/Problem 10
  ...circumcircle, so it has length <math>2 \cdot 3 = 6</math>. The triangle is similar to a 3-4-5 triangle with the ratio of their side lengths equal to <math>\fr [[Category:Introductory Geometry Problems]]
  2 KB (231 words) - 14:02, 3 June 2021
• 2006 AIME II Problems/Problem 6
  ...d AA~ we know that triangles <math>AA'D'</math> and <math>D'C'E</math> are similar. Thus, the sides are proportional: <math>\frac{AA'}{A'D'} = \frac{D'C'}{C'E ...olve in terms of the side <math>x</math> only (single-variable beauty)? By similar triangles we obtain that <math>BE=\frac{x}{1-x}</math>, therefore <math>CE=
  7 KB (1,067 words) - 12:23, 8 April 2024
• 2006 AIME II Problems/Problem 12
  Because the ratio of the area of two similar figures is the square of the ratio of the corresponding sides, <math>[GBC] ==Solution 2: Analytic Geometry/Coord Bash==
  6 KB (1,033 words) - 02:36, 19 March 2022
• Primary Mathematics World Contest
  ...similar to [[AMC 10]]/[[AMC 12]] level, with an emphasis on number theory, geometry and logic.
  1 KB (183 words) - 13:57, 15 October 2018
• 2004 AMC 12A Problems/Problem 18
  ...>\overline{AB}</math> is constructed inside the square, and the [[tangent (geometry)|tangent]] to the semicircle from <math>C</math> intersects side <math>\ove ...h>CDE</math> are in arithmetic progression. Thus it is [[similar triangles|similar]] to the triangle <math>3 - 4 - 5</math> and since <math>DC = 2</math>, <ma
  5 KB (738 words) - 13:11, 27 March 2023
• Lexington Math Tournament
  ...nd is replaced with an Accuracy Round, but the rest of the event follows a similar format. ...ach competition, but some examples of past mini-events are the estimathon, geometry bee, science bowl (or ball), and Tetris.
  3 KB (484 words) - 20:04, 12 March 2024
• Concurrence/Problems
  Hallie is teaching geometry to Warren. She tells him that the three medians, the three angle bisectors, To find the similar formula for <math>CE</math>, we just switch the signs of <math>BC^2</math>
  3 KB (458 words) - 15:44, 1 December 2015
• 2000 AMC 12 Problems/Problem 21
  ...e angle chasing shows that the two right triangles are [[similar triangles|similar]]. Thus the ratio of the sides of the triangles are the same. Since <math>A Since triangle <math>A</math> is similar to the large triangle, it has <math>h_A = a(\frac{c}{b}) = \frac{ac}{b}</ma
  5 KB (804 words) - 01:22, 13 May 2024
• 2007 AMC 10A Problems/Problem 21
  ...th> be the surface area of the inner square. The ratio of the areas of two similar figures is equal to the square of the ratio of their sides. As the diagonal [[Category:Introductory Geometry Problems]]
  4 KB (609 words) - 14:41, 3 December 2023
• 1951 AHSME Problems
  In connection with proof in geometry, indicate which one of the following statements is ''incorrect'': <math> \textbf{(A)}\ \text{the triangles are similar in opposite pairs}\qquad\textbf{(B)}\ \text{the triangles are congruent in
  23 KB (3,646 words) - 21:53, 21 June 2024
• 2002 AMC 12B Problems/Problem 20
  ...ht triangles <math>\triangle MON</math> and <math>\triangle XOY</math> are similar by Leg-Leg with a ratio of <math>\frac{1}{2}</math>, so <math>XY=2(MN)=\box [[Category:Introductory Geometry Problems]]
  3 KB (447 words) - 15:02, 17 August 2023
• 2002 AMC 12B Problems/Problem 25
  Similar to Solution 1, we proceed to get the area of the circle satisfying <math>f( [[Category:Introductory Geometry Problems]]
  2 KB (365 words) - 14:48, 7 March 2022

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