MathDB

R =2r/(r+1) , a square and two incircles

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2020 p2

March 26, 2024

geometrycirclessquare

Problem Statement

Consider a square

ABCD

. Let

M

be the midpoint of segment

AB

\Gamma_1

be the circle tangent to

\overline{AD}

\overline{AM}

and

\overline{MC}

with radius

r > 0

and let

\Gamma_2

be the circle tangent to

\overline{AD}

\overline{DC}

and

\overline{MC}

with radius

R > 0

. Prove that

R =\frac{2r}{r+1}