MathLabs

Problem 1

Let P be a point in the interior of triangle ABC. Let D, E, F be the intersections of AP with BC, BP with CA, and CP with AB, respectively. Prove that the area of triangle ABC is 6 if each of triangles PFA, PDB, and PEC has area 1.
Step 1 of 5: Name the three central areas
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x=[PAB],y=[PBC],z=[PCA],[PFA]=[PDB]=[PEC]=1.x=[PAB],\quad y=[PBC],\quad z=[PCA],\quad [PFA]=[PDB]=[PEC]=1.
Detailed analysis

Use [XYZ] for the area of triangle XYZ. The three triangles around P have areas x, y, z, while the three given outer pieces each have area 1.