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2012 AIME II Problems/Problem 15
...ive up after realizing that angle chasing won't work, you'd likely go in a similar approach to Solution 1 (below) or maybe be a bit more insightful and go wit [[Category:Intermediate Geometry Problems]]

13 KB (2,298 words) - 12:56, 10 September 2023
1988 USAMO Problems/Problem 4
...}</math>), so that <math>M</math> is the circumcenter of <math>AIC</math>. Similar results hold for <math>BIC</math> and <math>CIA</math>, and hence <math>O_c [[Category:Olympiad Geometry Problems]]

3 KB (504 words) - 19:25, 14 October 2021
1989 AHSME Problems/Problem 4
...m <math>A</math> and <math>B</math>; then the triangle <math>DBY</math> is similar to <math>DEF</math> but with corresponding sides of half the length. [[Category: Introductory Geometry Problems]]

2 KB (271 words) - 07:42, 22 October 2014
1978 USAMO Problems/Problem 2
.... In this case let AA' and BB' meet at O. Then triangles OAB and OA'B' are similar, so O must represent the same point. So assume A'B' is not parallel to AB. ...oordinate of O equals the y' coordinate of O. Similarly, XOX' and ZOZ' are similar, so OX/OZ = OX'/OZ', so the x-coordinate of O equals its x'-coordinate. In

4 KB (712 words) - 21:57, 12 November 2023
1978 USAMO Problems/Problem 4
...gles. Now, since <cmath>WX=2\cdot OW\sin\frac{\angle{WOX}}{2},</cmath> and similar for the other sides, we have that <math>WXYZ</math> is a regular tetrahedro [[Category:Olympiad Geometry Problems]]

2 KB (322 words) - 23:16, 18 July 2016
2013 AMC 12A Problems/Problem 9
...al triangles <math>\triangle BDE</math> and <math>\triangle EFC</math> are similar to <math>\triangle ABC</math>, and are therefore also isosceles triangles. [[Category:Introductory Geometry Problems]]

2 KB (352 words) - 16:25, 12 August 2023
2013 AMC 10A Problems/Problem 7
...ram of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least on Case <math>1</math>: The student chooses both algebra and geometry.

2 KB (379 words) - 13:06, 1 July 2023
2013 AMC 10A Problems/Problem 12
[[Category: Introductory Geometry Problems]] ...al triangles <math>\triangle BDE</math> and <math>\triangle EFC</math> are similar to <math>\triangle ABC</math>, and are therefore also isosceles triangles.

2 KB (376 words) - 11:09, 30 July 2022
2013 AMC 10B Problems/Problem 15
...hexagon can be broken into 6 small equilateral triangles, each of which is similar to the big equilateral triangle. The big triangle's area is 6 times the are [[Category:Introductory Geometry Problems]]

2 KB (240 words) - 12:18, 18 October 2022
2013 AMC 10B Problems/Problem 16
.../math>. Also, because <math>DE || AC</math>, <math>\triangle ABC</math> is similar to <math>\triangle{DBE}</math> with side length ratio <math>2:1</math>, so [[Category:Introductory Geometry Problems]]

5 KB (761 words) - 19:33, 11 January 2024
2013 AMC 12B Problems/Problem 13
...rithmetic progression. Triangles <math>ABD</math> and <math>DCB</math> are similar with <math>\angle DBA = \angle DCB</math> and <math>\angle ADB = \angle CBD For the sake of simplicity, lets rename the angles of each similar triangle. Let <math>\angle ADB = \angle CBD = \alpha</math>, <math>\angle D

3 KB (443 words) - 12:32, 8 January 2021
2013 AMC 12B Problems/Problem 19
[[Category: Introductory Geometry Problems]] ...}=\angle{ABF}</math>, so <math>\triangle ABF \sim \triangle ADE</math> are similar. In addition, <math>\triangle ADE \sim \triangle ACD</math>. We can easily

6 KB (1,004 words) - 22:38, 18 June 2023
2013 AMC 12B Problems/Problem 24
Furthermore, triangle <math>BCN</math> is similar to triangle <math>MCX</math>, so <math>BC/CM=CN/CX</math>, therefore <math> ...e have more similar triangles. In fact, going back to our original pair of similar triangles - <math>\triangle ANC</math> and <math>\triangle BXC</math> - giv

11 KB (1,876 words) - 00:08, 12 October 2023
1989 AHSME Problems/Problem 7
...can see that <math>\triangle ACH</math> and <math>\triangle MCN</math> are similar, implying <math>\overline{HN}=\overline{NC}</math>, implying that <math>\tr [[Category: Introductory Geometry Problems]]

2 KB (221 words) - 18:04, 21 October 2018
2014 AIME II Problems/Problem 11
Let <math>MN = x.</math> Meanwhile, since <math>\triangle R PM</math> is similar to <math>\triangle RCD</math> (angle, side, and side- <math>RP</math> and < [[Category:Intermediate Geometry Problems]]

6 KB (1,059 words) - 18:24, 20 January 2024
1967 AHSME Problems/Problem 20
Now we note that the picture is self-similar; if we erase the outer square, erase the outer circle, rotate the picture, [[Category:Introductory Geometry Problems]]

2 KB (390 words) - 01:40, 16 August 2023
Geometry Solutions
==Geometry Explained== This page is meant for explaining the most difficult problems in the geometry packet.

23 KB (3,182 words) - 12:30, 5 April 2014
1967 AHSME Problems/Problem 15
The difference in the areas of two similar triangles is <math>18</math> square feet, and the ratio of the larger area [[Category:Introductory Geometry Problems]]

1 KB (231 words) - 01:39, 16 August 2023
1967 IMO Problems/Problem 4
...o acute-angled triangles. Consider all triangles <math>ABC</math> that are similar to <math>\triangle A_1B_1C_1</math> (so that vertices <math>A_1</math>, <ma ...the triangles <math>ABC</math> circumscribed to <math>A_0B_0C_0</math> and similar to <math>A_1B_1C_1</math> by selecting <math>A</math> on <math>\mathcal C_A

2 KB (368 words) - 13:15, 29 January 2021
1962 AHSME Problems/Problem 16
...inches. Rectangle <math>R_2</math> with diagonal <math>15</math> inches is similar to <math>R_1</math>. Expressed in square inches the area of <math>R_2</math ...know that <math>\frac{b}a=\frac31=3</math>, because the two rectangles are similar. We also know that <math>a^2+b^2=15^2=225</math>. But <math>b=3a</math>, so

1 KB (170 words) - 22:16, 3 October 2014

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