Right Triangle Projection Theorem

Right-angled triangle projection theorem (also called Euclid's theorem): In a right-angled triangle, the height on the hypotenuse is the median of the ratio of the projections of the two right-angled sides on the hypotenuse. Each right-angled side is the median of the ratio between the projection of the right-angled side on the hypotenuse and the hypotenuse.

The formula is as shown in the figure. In Rt△ABC, ∠BAC=90°, AD is the height on the hypotenuse BC, then there is a projection theorem as follows:

1. (AD)^ 2=BD·DC,

2. (AB)^2=BD·BC,

3. (AC)^2=CD·BC.

This is mainly derived from similar triangles. For example, the proof of "(AD)^2=BD·DC:" is as follows:

In △BAD and △ACD, ∠B=∠DAC, ∠BDA=∠ADC=90°, △BAD∽△ACD is similar,

So AD/BD=CD/AD,

So (AD)^ 2=BD·DC.

Note: The above projection theorem can also prove the Pythagorean theorem. From formula (2) + (3), we get (AB)^2+ (AC)^2= (BC)^2. This is the conclusion of the Pythagorean theorem.