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• 2001 AIME II Problems/Problem 7
  ...STR \sim \triangle UQV</math>. Since the ratio of corresponding lengths of similar figures are the same, we have ...we find that <math>A_{1}Q = 60</math>, <math>A_{1}R = 90</math>. Using the similar triangles, <math>RA_{2} = 45</math>, <math>QA_{3} = 20</math>, so <math>A_{
  7 KB (1,112 words) - 02:15, 26 December 2022
• 2001 AIME II Problems/Problem 6
  ...hs are positive, we discard the other root). The ratio of the areas of two similar figures is the square of the ratio of their corresponding side lengths, so [[Category:Intermediate Geometry Problems]]
  4 KB (772 words) - 19:31, 6 December 2023
• 2000 AIME II Problems/Problem 6
  ...e segment into three partitions with lengths <math>x_1, 75, x_2</math>. By similar triangles, we easily find that <math>\frac{x - 75}{100} = \frac{x_1+x_2}{10 [[Category:Intermediate Geometry Problems]]
  3 KB (433 words) - 19:42, 20 December 2021
• 2005 AMC 10B Problems/Problem 14
  ...ve that <math>\triangle AYC</math> and <math>\bigtriangleup MXC</math> are similar. By the [[Pythagorean theorem]], <math>AY</math> is <math>\sqrt3</math>. [[Category:Introductory Geometry Problems]]
  5 KB (882 words) - 22:12, 30 April 2024
• 2005 IMO Shortlist Problems/G3
  ...ath>. It follows that triangles <math> \displaystyle ABK, LDA </math> are similar. Then since <math> \displaystyle ABCD </math> is a parallelogram, ...th>, this implies that triangles <math> \displaystyle KBC, CDL </math> are similar. This means that
  3 KB (453 words) - 10:53, 24 June 2007
• 1959 IMO Problems/Problem 5
  ...{MB} = \frac{AN}{NB} </math> since the triangles <math>ABN, BCM</math> are similar. Then <math>NM </math> bisects <math>ANB </math>. [[Category:Olympiad Geometry Problems]]
  4 KB (729 words) - 08:23, 23 May 2024
• Geometry/Introduction
  == Introductory Topics in Geometry == The following topics make a good introduction to [[geometry]].
  1 KB (122 words) - 16:25, 18 May 2021
• 1995 AHSME Problems/Problem 8
  <math>\triangle BDE</math> is similar to <math>\triangle BAC</math> by angle-angle similarity since <math>E=C = 9 [[Category:Introductory Geometry Problems]]
  1 KB (199 words) - 13:58, 5 July 2013
• Geometry/Olympiad
  An olympiad-level study of [[geometry]] involves familiarity with intermediate topics to a high level, a multitud === Synthetic geometry ===
  2 KB (242 words) - 10:16, 18 June 2023
• Equilateral triangle
  ...me angle measurements, so, by AA similarity, all equilateral triangles are similar). [[Category:Geometry]]
  1 KB (186 words) - 19:57, 15 September 2022
• Midpoint
  In [[Euclidean geometry]], the '''midpoint''' of a [[line segment]] is the [[point]] on the segment ...these segment lengths, <math>\Delta ABC \sim \Delta EFD (SSS)</math> with similar ratio 2:1. The area ratio is then 4:1; this tells us
  4 KB (596 words) - 17:09, 9 May 2024
• Volume
  ...</math>, where <math>(a_z,b_z)</math> are the bounds of <math>z</math> and similar bounds are defined for <math>x</math> and <math>y</math>. *[[Center]]s of adjacent [[face]]s of a unit [[cube (geometry) | cube]] are joined to form a regular [[octahedron]]. What is the [[volume
  3 KB (523 words) - 20:24, 17 August 2023
• Orthocenter
  ...2 GO</math>. Then the triangles <math>AGH</math>, <math>A'GO</math> are [[similar]] by side-angle-side similarity. It follows that <math>AH</math> is parall Euclidean Geometry in Mathematical Olympiads by Evan Chen - Section 1.3
  5 KB (829 words) - 13:11, 20 February 2024
• Similarity (geometry)
  ...tation. For example, a globe and the surface of the earth are, in theory, similar. ...tion, rotation and reflection ([[rigid motion]]s). We say two objects are similar if they are congruent up to a [[dilation]].
  2 KB (261 words) - 20:42, 25 November 2023
• 1970 IMO Problems/Problem 1
  By similar triangles and the fact that both centers lie on the angle bisector of <math [[Category:Olympiad Geometry Problems]]
  2 KB (380 words) - 22:12, 19 May 2015
• Euler line
  ...er [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distance [[ratio]]s hold among these points. I .... Consider the [[medial triangle]] <math>\triangle O_AO_BO_C</math>. It is similar to <math>\triangle ABC</math>. Specifically, a rotation of <math>180^\circ<
  59 KB (10,203 words) - 04:47, 30 August 2023
• Nine-point circle
  That such a circle exists is a non-trivial theorem of Euclidean [[geometry]]. ...h>O_bE_b</math> and <math>O_cE_c</math> are diagonals of the circumcircle. Similar logic to the above gives us that <math>O_aO_cE_aE_c</math> is a rectangle w
  6 KB (994 words) - 16:02, 12 March 2024
• Pick's Theorem
  It is similar to the [[Shoelace Theorem]], and although it is less powerful, it is a good [[Category:Geometry]]
  2 KB (301 words) - 13:08, 20 February 2024
• 2003 AMC 10A Problems/Problem 22
  Therefore <math>\triangle GFA</math> and <math>\triangle ABH</math> are similar. <math>\triangle GCH</math> and <math>\triangle GEA</math> are also similar.
  9 KB (1,446 words) - 22:48, 8 May 2024
• 1987 IMO Problems/Problem 2
  ...ular. Furthermore, we have triangles <math>ABN</math> and <math>ALC</math> similar because two corresponding angles are equal. [[Category:Olympiad Geometry Problems]]
  4 KB (736 words) - 15:39, 21 September 2014

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