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cos (AB,OM) wanted if sum of areas (AMC)+( BMD) is min

Source: Spanish Mathematical Olympiad 1970 P8

December 4, 2022

geometrygeometric inequality

Problem Statement

There is a point

M

inside a circle, at a distance

OM = d

of the center

O

. Two chords

AB

and

CD

are traced through

M

that form a right angle . Join

A

with

C

and

B

with

D

. Determine the cosine of the angle that must form the chord

AB

with

OM

so that the sum of the areas of the triangles

AMC

and

BMD

be minimal.