In each ratio of two segments along the altitude, the ratio is the same as the ratio of the areas of two triangles to the same base.

The ratio of the areas of triangles HBC and ABC is the same as the ratio HD/HD. Hence, considering all three cases,

But, the products, AD.BC, BE.AC, and CF.AD, are all the same, each being twice the area of the triangle ABC. Thus they could be written over a common denominator K = twice the area.

Interpret each of the three products. Each is twice the area of one of the three subtriangles with vertex at H and a base along once side of triangle ABC. There for the sum of these products is K and the result for the first equation is obtained.