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Regional Olympiad - FBH 2016 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

September 22, 2018

circlegeometry

Problem Statement

Circle of radius

R_1

is inscribed in an acute angle

\alpha

. Second circle with radius

R_2

touches one of the sides forming the angle

\alpha

in same point as first circle and intersects the second side in points

A

and

B

, such that centers of both circles lie inside angle

\alpha

. Prove that

AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}