To prove Triangle APB is similar to Triangle APC
First note that Angle PAB is a right angle because the tangent PA must be perpendicular to the diameter AB at the point of contact.
Also angle ACB is a right angle because it is an angle in a semi -circle, so Angle ACP =180-90 =90 degrees
(ACP is a straight angle)
Hence Considering Triangle APB and Triangle APC
Angle PAB = Angle ACP= 90 degrees as shown above
Angle APB =Angle APC (same angle)
Angle PBA = Angle CAP ( Supplementary to 2 equal angles in the triangles)
Hence since the two triangles are equangular, they must be similar.
Triangle APB is similar to Triangle APC
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