Problem
GEO-B2-M07-P024 Areas Determine All Three Cevians
#24
★★★★★ Level 5 of 5
Inside triangle \(ABC\), point \(P\) is to be chosen so that \([PBC]:[PCA]:[PAB]=6:10:15\). If \(AP\), \(BP\), \(CP\) meet the sides at \(D,E,F\), find \(BD:DC\), \(CE:EA\), \(AF:FB\), and check that these ratios agree with Ceva.
Use \(S_A=6\), \(S_B=10\), \(S_C=15\).
By the formulas, \(\frac{BD}{DC}=\frac{S_C}{S_B}=\frac{15}{10}=3:2\). Next, \(\frac{CE}{EA}=\frac{S_A}{S_C}=\frac{6}{15}=2:5\). Finally, \(\frac{AF}{FB}=\frac{S_B}{S_A}=\frac{10}{6}=5:3\). Ceva check: \(\frac{3}{2}\cdot\frac{2}{5}\cdot\frac{5}{3}=1\), so the ratios are consistent with concurrence of the cevians.
The final problem reverses the usual logic: first areas, then cevians.