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0213 geometry 2nd edition Round 1 p3

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May 10, 2021
geometry2nd edition

Problem Statement

Given are on a line three points A,B,CA, B, C such that AB=1AB = 1 and BC=xBC = x. Consider the circles Ωa,Ωb\Omega_a, \Omega_b and Ωc\Omega_c which are tangent to the given line at the points A,B,CA, B, C respectively, and such that Ωb\Omega_b is tangent externally with both Ωa\Omega_a and Ωc\Omega_c in points M,NM, N respectively. Find all values of the radius of the circle Ωb\Omega_b for which the triangle BMNBMN is isosceles.
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