tag:blogger.com,1999:blog-2400513859305780710.post2810375160900954311..comments2023-04-04T07:53:53.789-07:00Comments on Mathematical Food For Thought: I See Double! Topic: Geometry. Level: AIME/Olympiad.Jeffrey Wanghttp://www.blogger.com/profile/11114458640271201663noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-2400513859305780710.post-72989180306599930232006-03-16T10:54:16.000-08:002006-03-16T10:54:16.000-08:00Wow, very nice problem and solution.Wow, very nice problem and solution.DPopovnoreply@blogger.comtag:blogger.com,1999:blog-2400513859305780710.post-88070616178777850632006-03-17T18:14:46.000-08:002006-03-17T18:14:46.000-08:00In triangle ABC draw the circumcenter P. We have ...In triangle ABC draw the circumcenter P. We have PA = PB = PC. Now consider the triangles PAB, PBC, PAC. Knowing that angle BPC is twice angle BAC, etc., we infer that triangle PAB is isosceles with angle APB = 2A, etc. <br><br>Now, drop altitudes from P. We have three pairs of right triangles with angles A, B, C, and from then we derive the relations<br><br>sin A = BC / (2R)<br>sin B = AC / (2R)<br>sin C = AB / (2R)<br><br>Which rearranges to form the Extended Law of Sines. (This relies on a property of circles I cannot prove, unfortunately. =/ )QCnoreply@blogger.com