Point D is in the interior of △ABC. Line CD intersects AB at point M. Prove that, if AADC=ABDC, then CM is the median of △ABC.

Question

Point D is in the interior of △ABC. Line CD intersects 

AB

  at point M. Prove that, if A_ADC=A_BDC, then 

CM

  is the median of △ABC.    

Answer 1

Area = 1/2 * base * height

The areas are the same

Then for height CM,

AD* CM = BD * CM

which forces

AD = BD

Right triangles ADM and BDM are congruent

So then AM = BM

which forces CM to be the median