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max sum CM + ON, square on side of a triangle

Source: TOT 374 1993 Spring A S2 - Tournament of Towns

June 10, 2024

geometrygeometric inequality

Problem Statement

A square is constructed on the side

AB

of triangle

ABC

(outside the triangle).

O

is the centre of the square.

M

and

N

are the midpoints of the sides

BC

and

AC

. The lengths of these sides are

a

and

b

respectively. Find the maximal possible value of the sum

CM + ON

(when the angle at

C

changes).
(IF Sharygin)