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• 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 (505 words) - 00:51, 26 July 2024
• 2002 AMC 10A Problems/Problem 13
  ...is dropped (the big triangle and two smaller ones it gets split into) are similar! We take the triangle with hypotenuse <math>20</math>. The ratio of sides b [[Category:Introductory Geometry Problems]]
  3 KB (395 words) - 13:22, 8 November 2021
• 2002 AMC 10A Problems/Problem 20
  ...l, triangles <math>\triangle GAD</math> and <math>\triangle HCD</math> are similar. Hence, <math>\frac{CH}{AG} = \frac{CD}{AD} = \frac{1}{3}</math>. ...l, triangles <math>\triangle GAF</math> and <math>\triangle JEF</math> are similar. Hence, <math>\frac{EJ}{AG} = \frac{EF}{AF} = \frac{1}{5}</math>. Therefore
  3 KB (439 words) - 22:15, 9 June 2023
• 1972 USAMO Problems/Problem 5
  ...orresponding vertices. From the lemma, pentagon <math>A'B'C'D'E'</math> is similar to pentagon <math>ABCDE</math> with <math>AB = m(A'B')</math> and parallelo ...lograms, <math>BD' = EC' = (m - 1)a</math>. Triangle <math>EB'C'</math> is similar to triangle <math>ECB</math>, so <math>me = BC = \left(\frac{2m - 1}{m - 1}
  4 KB (684 words) - 09:29, 26 March 2023
• 1960 IMO Problems/Problem 7
  ...h>h - x</math> must be the distance from <math>P</math> to the other base. Similar triangles give <math>\frac{x}{\frac{a}{2}} = \frac{\frac{c}{2}}{h - x}</mat [[Category:Olympiad Geometry Problems]]
  2 KB (410 words) - 15:25, 23 March 2020
• 2000 AMC 10 Problems/Problem 5
  (a) Triangles <math>ABP</math> and <math>MNP</math> are similar, and since <math>PM=\frac{1}{2}AP</math>, <math>MN=\frac{1}{2}AB</math>. [[Category:Introductory Geometry Problems]]
  2 KB (260 words) - 00:38, 15 July 2024
• 2000 AMC 10 Problems/Problem 16
  Extend line <math>\overline{DC}</math> as above. This creates two similar triangles whose side lengths have the ratio <math>5:4</math>. Therefore <m [[Category:Introductory Geometry Problems]]
  7 KB (966 words) - 23:22, 31 July 2023
• 1995 USAMO Problems/Problem 3
  ...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</ma [[Category:Olympiad Geometry Problems]]
  2 KB (364 words) - 01:42, 19 April 2024
• 2004 AMC 10B Problems/Problem 18
  ...he triangles <math>\triangle AFG</math> and <math>\triangle AEC</math> are similar, as they have all angles equal. Their ratio is <math>\frac {AF}{AE} = \frac ==Solution 3 (Coordinate Geometry)==
  6 KB (904 words) - 12:54, 22 October 2023
• 2009 AMC 12A Problems/Problem 20
  [[Category: Introductory Geometry Problems]] ...B = \angle DEC</math>, triangles <math>AEB</math> and <math>DEC</math> are similar. Their ratio is <math>\frac {AB}{CD} = \frac {3}{4}</math>. Since <math>AE
  3 KB (541 words) - 15:56, 1 August 2024
• 2009 AMC 10A Problems/Problem 17
  [[Category: Introductory Geometry Problems]] Triangle <math>EAB</math> is similar to <math>BAD</math>, as they have the same angles. Segment <math>BA</math>
  4 KB (684 words) - 21:14, 23 October 2023

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