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for no point M 2 triangles and a rectangle have the same area

Source: 1st Mexican Mathematical Olympiad 1987 OMM P5

July 27, 2018

geometryrectangleareasright triangle

Problem Statement

In a right triangle

ABC

, M is a point on the hypotenuse

BC

and

P

and

Q

the projections of

M

AB

and

AC

respectively. Prove that for no such point

M

do the triangles

BPM, MQC

and the rectangle

AQMP

have the same area.