Circle Proof

[SmokeDawg]

Having some problems with this one, can anyone provide the answer? Only thing I have so far is:
Angle ACB = 90 by Thales' theorem and angle BAP = 90 because AP is a tangent.

 

[soroban]

Hello, SmokeDawg!

You have all the necessary facts . . .

Having some problems with this one, can anyone provide the answer?
Only thing I have so far is: angle ACB = 90^o by Thales' theorem
and angle BAP = 90^o because AP is a tangent.

Consider right triangles \(\displaystyle ACP\) and \(\displaystyle BAP.\)

We have: .\(\displaystyle \angle ACP \,=\,\angle BAP \,=\,90^o\)
. . . Also: .\(\displaystyle \angle P \,=\,\angle P\)

Therefore: .\(\displaystyle \Delta ACP \sim \Delta BAP \;\;\text{ (a.a.a.)}\)