As in the figure, let us denote by $u,v,w,x,y,z$ the areas of the six component triangles. Given any two triangles of the same height, their areas are in the same proportion as their bases (Euclid VI.1). Therefore

and the conclusion we want is

Clearing the denominators, the hypotheses are

which imply

and the conclusion says that

equals

or equivalently (after cancelling the underlined terms)

equals

i.e.

i.e. by (1)

i.e. by (3)

Using (4), we are down to

i.e. by (3)

i.e.

But in view of (2), this is the same as (4), and the proof is complete.

Remarks: Ceva’s theorem is an easy consequence of (4).