Πέμπτη, 12 Οκτωβρίου 2017

1999 JBMO problem 4 (GRE)

proposed by Greece

Let ${ABC }$  be a triangle with ${AB = AC }$. Also, let ${D \in [BC] }$  be a point such that ${BC > BD >DC > 0 }$, and let ${C_1, C_2 }$  be the circumcircles of the triangles ${ABD }$  and ${ADC }$  respectively. Let ${BB' }$  and ${CC' }$ be diameters in the two circles, and let ${M }$  be the midpoint of ${B'C'}$. Prove that the area of the triangle ${MBC }$  is constant (i.e. it does not depend on the choice of the point ${D }$).

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