Problem
GEO-B2-M07-P018 Proving Concurrence by Areas
#18
★★★★☆ Level 4 of 5
In triangle \(ABC\), points \(D,E,F\) lie on \(BC,CA,AB\). It is known that there exist positive numbers \(x,y,z\) such that \(\frac{BD}{DC}=\frac{z}{y}\), \(\frac{CE}{EA}=\frac{x}{z}\), \(\frac{AF}{FB}=\frac{y}{x}\). Prove that \(AD\), \(BE\), \(CF\) are concurrent.
Multiply the three ratios.
The product is \(\frac{z}{y}\cdot\frac{x}{z}\cdot\frac{y}{x}=1\). By Ceva's theorem, lines \(AD\), \(BE\), \(CF\) are concurrent. The numbers \(x,y,z\) may be interpreted as the areas of the three small triangles around the intersection point.
An abstract form of areas around a point.