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• 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
• 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

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