Online Geometry theorems, problems, solutions, and related topics.
Proposed ProblemSee also:Complete Problem 27, Collection of Geometry ProblemsLevel: High School, SAT Prep, College geometry
Hints: Two tangent segments to a circle are congruent. Square... See also Problem 668
http://img98.imageshack.us/img98/3497/problem27.pngConnect CO and AONote that CO and CO2 are angle bisector of angle (BCA) soC, O, O2 are collinearTriangle OGC congruence with triangle OKC ( right triangles with common hypotenuse and CO is angle bisector)So OC is the angle bisector of angle (GOK) >>Incircle O2 tangent to OG also tangent to OK and O2N=r2Similarly AO is angle bisector of angle HOK) and incircle O1 tangent to both OH and OK and O1P=r1DE=DK+KE=O1P+O2N=r1+r2Peter Tran
Using Antonio's hint above....If X is the point of tangency of circle O with AC AK = c - r and KC = a - rSimilarly AD = c-r-r1 and CE = a-r-r2So AC - AD - EC = r1+r2 + (b-c-a + 2r) from using the above expressions for AD and CE.The expression within the bracket = 0 and the result followsSumith PeirisMoratuwaSri Lanka