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• 2007 AMC 10A Problems/Problem 18
  ...at triangles <math>\triangle AMH</math> and <math>\triangle GMC</math> are similar because <math>\overline{AH} \parallel \overline{CG}</math>. Also, since <ma The triangles <math>CBN</math> and <math>CGH</math> are clearly similar with ratio <math>1:2</math>, hence <math>BN=2</math> and thus <math>AN=6</m
  6 KB (867 words) - 00:17, 20 May 2023
• 2004 AMC 12B Problems/Problem 19
  Above, <math>E,F,</math> and <math>G</math> are points of [[tangent (geometry)|tangency]]. By the Two Tangent Theorem, <math>BF = BE = 18</math> and <mat ...erline{FH} = 3x</math>.Triangles <math>OFH</math> and <math>BEH</math> are similar so <math>\frac{\overline{OF}}{\overline{BE}} = \frac{\overline{FH}}{\overl
  3 KB (520 words) - 19:12, 20 November 2023
• 2007 AMC 10B Problems/Problem 11
  <math>\triangle ADC</math> is similar to <math>\triangle ACR</math>, so <cmath>\frac {2\sqrt{2}}{3} = \frac {3}{2 [[Category:Introductory Geometry Problems]]
  5 KB (851 words) - 22:02, 26 July 2021
• 2002 AMC 10A Problems/Problem 25
  ...F</math> is similar to triangle <math>BEC</math>, we can use properties of similar triangles and find that <math>DE = 12 \cdot \frac{5}{13} = \frac{60}{13}</m [[Category:Introductory Geometry Problems]]
  8 KB (1,308 words) - 07:05, 19 December 2022
• Isogonal conjugate
  One can prove a similar theorem in the case <math>P</math> outside <math>\triangle ABC.</math> ...cated on the ray <math>OA_1</math> so that <math>\triangle OAA_1</math> is similar to <math>\triangle OA_2A</math>. Points <math>B_2</math> and <math>C_2</mat
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 12B Problems/Problem 11
  ...c 1 8</math> that of the entire cone (mountain). These cones are obviously similar so the radius and height of the small cone must be <math>\sqrt[3]{\frac1 8} [[Category:Introductory Geometry Problems]]
  2 KB (251 words) - 22:12, 8 May 2021
• 2008 AIME I Problems/Problem 14
  We let <math>AC=x</math>. From similar triangles, we have that <math>PC=\frac{x\sqrt{x^2+18x}}{x+9}</math> (Use Py ...nt on <math>x</math> is that it must be greater than <math>0</math>. Using similar triangles, we can deduce that <math>PA=\frac{9x}{x+9}</math>. Now, apply la
  8 KB (1,333 words) - 00:18, 1 February 2024
• 2008 AIME II Problems/Problem 5
  Similar to solution 1; Notice that it forms a right triangle. Remembering that the [[Category:Intermediate Geometry Problems]]
  8 KB (1,338 words) - 23:15, 28 November 2023
• 2008 AIME II Problems/Problem 11
  ...eta) = \frac {3}{4}</math>. Therefore, <math>XC = \frac {64}{3}</math>. By similar reasoning in triangle <math>QBY</math>, we see that <math>BY = \frac {4r}{3 ...- (\overline{PZ})^2} = \sqrt{(16+r)^2-(16-r)^2} = 8\sqrt{r}</cmath> Using similar similarity as was done to find <math>\overline{BY}</math> we have <math>\fr
  6 KB (1,065 words) - 20:12, 9 August 2022
• 1998 APMO Problems/Problem 4
  ...ath> are similar triangles, so <math>AFEN</math> and <math>ACBM</math> are similar figures. It follows that [[Category:Olympiad Geometry Problems]]
  2 KB (345 words) - 18:42, 13 April 2008
• 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

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