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 3 of 5: Cycle the same argument
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(x+1)y=x+y+z,(y+1)z=x+y+z,(x+1)y=(y+1)z=(z+1)x.(x+1)y=x+y+z,\qquad (y+1)z=x+y+z,\qquad (x+1)y=(y+1)z=(z+1)x.
Detailed analysis

Applying the identical area-ratio argument to the cevians through A and B gives the other two equations. Hence the three products in the last equality are equal.