Problem

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 $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.