Points M, N, P lie on sides AB, BC and CA of triangle ABC, and AM: AB = BN: BC

Points M, N, P lie on sides AB, BC and CA of triangle ABC, and AM: AB = BN: BC = CP: CA = 1: 3. The area of triangle MNP is S. Find the area of triangle ABC.

Let us express the areas of the triangles AMR, ВMН, CHP through the area of the triangle ABC.

Saavs = AB * AC * CosA.

Samr = AM * AC * CosA = (AB / 3) * (2 * AC / 3) * CosA = (2/9) * AB * BC * CosA.

Then Saavs / Samr = 2/9.

Samr = 2 * Savs / 9.

Similarly:

Svmn = 2 * Savs / 9.

Scnr = 2 * Saavs / 9.

Then Savs = Samr + Svmn + Ssnr + Smnr = (2 * Savs / 9) + (2 * Savs / 9) + (2 * Savs / 9) + S.

Savs = (2 * Savs / 3) + S

Savs – (2 * Savs / 3) = S.

Savs / 3 = S.

Sас = 3 * S.

Answer: The area of the triangle ABC is 3 * S cm2.