Proposed Problem
See complete Problem 279 at:
gogeometry.com/problem/p279_tangent_circle_common_external_tangent_chord_inradius.htm
Level: High School, SAT Prep, College geometry
Thursday, April 2, 2009
Problem 279: Tangent Circles, Common External Tangent, Chords, Inradius
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Using results from earlier problems, we know that: CD^2 = 4ba^2/(a+b), CE^2=4ab^2/(a+b) and DE^2=4ab. Further, r=CD*CE/(CD+CE+DE)
ReplyDeleteThus,r=(2aVb*2bVa)/(2aV(b/(a+b))+2aV(a/(a+b))+2V(ab)) where V = square root
Thus, r = [4abV(ab/(a+b))]/2aVb+2bVa+2V(ab(a+b)
Upon rationalization & simplification it can easily be show that:
r=Vab[Va+Vb-V(a+b)]/V(a+b) or
r=V(ab)[(Va+Vb)V(a+b) -1]
QED
Ajit: ajitathatle