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two diameters of circle, points concyclic (Bulgaria 1962 P2)

Source:

June 24, 2021

geometrycircles

Problem Statement

It is given a circle with center

O

and radius

r

.

AB

and

MN

are two diameters. The lines

MB

and

N''

are tangent to the circle at the points

M'

and

N'

and intersect at point

A

.

M''

and

N''

are the midpoints of the segments

AM'

and

AN'

. Prove that:
(a) the points

M,N,N',M'

are concyclic. (b) the heights of the triangle

M''N''B

intersect in the midpoint of the radius

OA

.

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