Height BD is drawn in right-angled triangle ABC from right angle B. Prove that BD = AB * BC / AC.

First way.

In a right-angled triangle ABC, we define its area in two ways, through the lengths of the legs and through the base and height.

Savs = AB * BC / 2.

Savs = AC * ВD / 2.

Equate the right sides of the equations and multiply them by 2.

2 * AB * BC / 2 = 2 * AC * ВD / 2.

AB * BC = AC * ВD.

ВD = AB * BC / AC.

Second way.

In right-angled triangles ABC and ВСD, we express the sine of the angle C.

SinC = AB / AC.

SinC = СD / BC.

Then AB / AC = СD / BC.

СD = AB * BC / AC.

Q.E.D.