Prove triangle is isosceles

Shape and space, angle and circle properties, ...
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Holbi2004
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Joined: Tue May 01, 2018 7:01 pm

Prove triangle is isosceles

Post by Holbi2004 » Sun May 13, 2018 8:04 pm

Let AE, BF and CG be the angle bisectors of triangle ABC. Let the points E, F and G lie on BC, AC and AB, respectively. Let M denote the incenter of ABC.

Prove that if the sum of the areas of the inside triangles MCF, MAG and MBG equals half the area of triangle ABC, then ABC is isosceles. In otrher words:

area(MCF) + area(MAG) + area(MBG) = 1/2 * area (ABC)

=> ABC is isosceles. Will Ceva´s theorem and the angle bisector theoerem bring me any further?

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