Inside triangle ABC, point M is taken such that the areas of triangles AMB, BMC

Inside triangle ABC, point M is taken such that the areas of triangles AMB, BMC and AMC are equal. Prove that M is the intersection of the medians of this triangle.

From point M we draw medians MK, MH, MP of triangles AMB, CMB, AMB.

By the property of the medians of triangles, the area of triangles is AMK = BMK, BMH = CMH, AMP = CMP.

Savr = Samr + Savm.

Svsr = Smr + S imn.

Then the area of the triangles ABP = BCP. Then BP is the median ABC.

Similarly, the area of the triangle BMH = CMH, and AH is the median ABC.

Then the point M is the point of intersection of the medians, as required.