Geometry-The Euclid Theorem Part1

The Euclid Theorem, also known as  射影定理, can be used when there's an triangle like this (Not really): 

                                                                                          

As can be seen, m∠ABC=90° and BD⊥AC

The following statements are true based on the given information.

1. (BD)^2=AD*DC，2.(AB)^2=AD*AC ，3.(BC)^2=CD*CA , 4.AB*BC=AC*BD

Proof for the the first statement: 

Since m∠BDC=90

m∠DBC+m∠C=90 (corollary to the triangle sum theorem)

and SInce m∠ABC=90

m∠A+m∠C=90   (corollary to the triangle sum theorem)

m∠DBC=m∠A

for the same reason

m∠A+m∠ABD=90

m∠A+m∠C=90

m∠C=m∠ABD

Since ∠C≌∠ABD and m∠BDA=m∠BDC=90

By the Angle-Angle Similarity Postulate

△ABD∽△BCD

Therefore AD/BD=BD/CD

(BD)^2=AD*DC