Consider a triangle ABC and a point P in its interior. Lines PA, PB, PC intersect BC, CA, AB at A′,B′,C′ , respectively. Prove that BA′BC+CB′CA+AC′AB=32if and only if at least two of the triangles PAB, PBC, PCA have the same area.
Source: 2010 Saudi Arabia BMO TST 3.2 - Balkan
Tags: ratio, equal areas, geometry, Cevians
Consider a triangle ABC and a point P in its interior. Lines PA, PB, PC intersect BC, CA, AB at A′,B′,C′ , respectively. Prove that BA′BC+CB′CA+AC′AB=32if and only if at least two of the triangles PAB, PBC, PCA have the same area.