Construct a tangent line from an external point to a circle.
Identify the tangent points in the auxiliary-circle construction
Problem
An auxiliary circle with diameter \(OP\) intersects the original circle at \(A\) and \(B\). Name the \(\text{two}~\text{points}\) that serve as the tangency points for the tangent segments drawn from \(P\).
The two-circle construction hides the tangent condition inside a diameter: each shared intersection subtends the auxiliary diameter as a right angle. We’ll identify those intersections, pair each with a radius from the original center, and use perpendicularity to decide which points are the contacts.
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