Problem
GEO-B1-M08-P026 Area Form of Ceva
#26
★★★★★ Level 5 of 5
In triangle \(ABC\), lines \(AD\), \(BE\), \(CF\) meet at one point \(P\), where \(D\) lies on \(BC\), \(E\) on \(CA\), and \(F\) on \(AB\). Prove that \(\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}=1\).
Express segment ratios through area ratios of triangles with vertex \(P\).
Through areas, \(\frac{BD}{DC}=\frac{S_{ABP}}{S_{ACP}}\), \(\frac{CE}{EA}=\frac{S_{BCP}}{S_{ABP}}\), and \(\frac{AF}{FB}=\frac{S_{ACP}}{S_{BCP}}\). Multiplying the three equalities, all areas cancel, and the result is \(1\).
A challenge problem: a bridge to Book 2, but solved only by areas.