In the right-angled triangle ADC from the vertex of the right angle C

In the right-angled triangle ADC from the vertex of the right angle C, the perpendicular CD to the hypotenuse is drawn. Prove that BC ^ 2 = BD * BA

Let us prove that triangles ABC and CBD are similar.

Both triangles are rectangular. Let the angle A = X0, then the angle ACB is equal to (90 – X).

In the triangle CBD, the angle BCD = (90 – (90 – X) = X0.

Triangles ABC and CBD are similar in acute angle.

Then the ratio of the sides of such triangles is:

BC / ВA = BD / BC.

ВС ^ 2 = ВD * ВA.

Q.E.D.