Construct a tangent line from an external point to a circle.
Construct tangents with the midpoint auxiliary circle
Problem
Construct \(\text{both}~\text{tangents}\) from exterior point \(P\) to the circle centered at \(O\). Give the ordered steps for bisecting \(OP\) at \(M\), drawing the diameter-based auxiliary circle, naming its intersections \(T\) and \(U\) with the original circle, and drawing the \(\text{two}~\text{tangent}~\text{segments}\).
The midpoint auxiliary circle is a device for manufacturing right angles, not just an extra circle. We’ll make the center-to-external-point segment a true diameter, use the two circles’ intersections to force perpendicular radius-segment pairs, and only then certify the constructed lines.
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