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• 1995 AHSME Problems/Problem 19
  ...e smaller triangle is <math>\sqrt{3}</math>. The ratio of the areas of two similar triangles is the square of the ratio of two corresponding side lengths, so [[Category:Introductory Geometry Problems]]
  1 KB (200 words) - 20:58, 10 February 2019
• Shoelace Theorem
  ...th>(x_1, y_1, z_1), (x_2, y_2, z_2),(x_3, y_3, z_3),(0,0,0)</math>. Does a similar formula hold for <math>n</math>Dimensional triangles for any <math>n</math> [[Category:Geometry]]
  8 KB (1,358 words) - 15:32, 22 February 2024
• 2008 AMC 10A Problems/Problem 18
  This solution is very similar to Solution 1, except instead of subtracting <math>a+b</math> from both sid [[Category:Introductory Geometry Problems]]
  8 KB (1,339 words) - 14:15, 1 August 2022
• 2008 USAMO Problems/Problem 2
  '''Lemma.''' <math>\triangle FEO</math> is directly similar to <math>\triangle NEM</math> ...rity. It is easy to see that they are oriented such that they are directly similar.
  20 KB (3,565 words) - 11:54, 1 May 2024
• 2003 AMC 12A Problems/Problem 17
  ...find <math>XJ</math> and double it to get our answer. With some analytical geometry, we find that <math>XJ=\frac{8}{5}</math>, implying that <math>PK=\frac{16} Manipulating similar triangles gives us <math>PZ=\frac{16}{5}</math>
  6 KB (1,026 words) - 22:35, 29 March 2023
• 1995 AHSME Problems/Problem 26
  ...triangle CFD</math> and <math>\triangle EOD</math> are [[Similar(geometry)|similar]]. [[Category:Introductory Geometry Problems]]
  2 KB (359 words) - 20:01, 23 January 2017
• How should I prepare?
  ...re for math contests. They cover a broad range of topics, from algebra to geometry to number theory to combinatorics and much much more. * Introduction to Geometry - [[Mathcounts]], [[AMC 8]], [[AMC 10]], [[AMC 12]], [[AIME]], [[HMMT]]
  13 KB (1,895 words) - 21:26, 10 June 2024
• 2003 USAMO Problems/Problem 4
  ...</math> if and only if triangles <math>BCM</math> and <math>CDM</math> are similar, that is ...D = \angle BMC</math>, triangles <math>CMD</math> and <math>BMC</math> are similar. Consequently, <math>\angle MCD = \angle MBC</math>. This exact condition c
  5 KB (820 words) - 02:39, 10 January 2023
• 2000 USAMO Problems/Problem 2
  ...espectively. Prove that all triangles in <math>S</math> are isosceles and similar to one another. ...e proved that all possible ABC are isosceles (as b = c), and that they are similar to a 5-5-8 triangle.
  4 KB (703 words) - 18:40, 3 January 2019
• 2001 IMO Shortlist Problems/G1
  ...ther is on <math>AC</math>. Points <math>B_1,\ C_1</math> are defined in a similar way for inscribed squares with two vertices on sides <math>AC</math> and <m ...mplies that quadrilaterals <math>AEA_1F</math> and <math>ABA_2C</math> are similar. Therefore <math>\angle BAA_2=\angle EAA_1</math> and <math>\angle CAA_2=EA
  3 KB (584 words) - 15:07, 12 December 2011
• 2001 IMO Shortlist Problems/G3
  By similar reasoning, we can calculate <math>BG^2=(2c^2+2a^2-b^2)/9</math> and <math>C [[Category:Olympiad Geometry Problems]]
  2 KB (430 words) - 18:13, 29 January 2019
• Millennium Problems
  ...ort of mathematical research. However, specifically, the inspiration was a similar prize exactly a hundred years earlier. Paris had seen a similar event then, at the second International Congress of Mathematicians. The fam
  13 KB (1,969 words) - 17:57, 22 February 2024
• 1984 IMO Problems/Problem 4
  ...D</math>. Then, since triangles <math>MNE</math> and <math>BCE</math> are similar, we have <math>\frac{CH}{r}=\frac{BG}{\frac{1}{2}AB}</math>. This gives an ...{\sin ECB}=\frac{EC}{EB}</math>. Since triangles <math>EDA,ECB</math> are similar, we have <math>\frac{EC}{EB}=\frac{CD}{AB}</math> and thus we have
  4 KB (750 words) - 23:49, 29 January 2021
• Carnot's Theorem
  ...ication of pythogorean theorem to triangles <math>PXB,PXC</math>. Now with similar relations for <math>Y</math> and <math>Z</math>, Carnot's theorem finishes [[Category:Geometry]]
  4 KB (723 words) - 01:45, 18 February 2021
• 2000 USAMO Problems
  ...espectively. Prove that all triangles in <math>S</math> are isosceles and similar to one another. ...>k = 2, 3, \dots, 7,</math> <math>\omega_k</math> is externally [[tangent (geometry)|tangent]] to <math>\omega_{k - 1}</math> and passes through <math>A_k</mat
  3 KB (495 words) - 19:02, 18 April 2014
• 2008 iTest Problems
  ...ua asks, "The battery shapes never change? Each year the new batteries are similar in shape - in all dimensions - to the batteries from previous years?" ...r she chooses <math>4</math> or <math>5</math> students at a time to tutor geometry each week:
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2008 iTest Problems/Problem 99
  ...ngle, we see that <math>[A_1A_{11}A_{12}] = \frac{1}{9}[AA_1A_2]</math>. A similar clipping about <math>A_2</math> gives <math>[A_2A_{21}A_{22}] = \frac{1}{9} [[Category:Olympiad Geometry Problems]]
  4 KB (685 words) - 14:39, 7 October 2017
• 2007 AMC 12B Problems/Problem 25
  Similar to solution 1, we allow [[Category:Introductory Geometry Problems]]
  3 KB (470 words) - 19:46, 17 July 2023
• 1999 AHSME Problems/Problem 2
  <math>\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.} </math> ...es in an equilateral triangle are congruent, all equilateral triangles are similar by AAA similarity. Thus, <math>E</math> is true.
  2 KB (259 words) - 14:28, 13 February 2019
• 2002 AMC 12A Problems/Problem 7
  ...mplify to get <math>\frac{r_1}{r_2}=\frac{2}{3}</math>. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of [[Category:Introductory Geometry Problems]]
  3 KB (492 words) - 14:46, 31 January 2024

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