Construct a tangent line from an external point to a circle.
Verify an external tangent with a semicircle right angle
Problem
A circle centered at \(O\) has \(OT~=~6\) and \(OP~=~10\). The auxiliary circle with diameter \(OP\) meets the original circle at \(T\). Use the angle subtending diameter \(OP\) to prove \(PT\) is tangent at \(T\), then calculate \(PT\) with the right triangle \(OTP\).
The auxiliary diameter does two jobs: it proves a right angle at the circle intersection and turns the radius, center distance, and tangent segment into a right triangle. We’ll establish tangency before doing arithmetic, identify the hypotenuse, and take the positive root required for a length.
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