Construct a tangent line from an external point to a circle.
Audit an external-tangent construction
Problem
For circle center \(O\) and external point \(P\), a proposed tangent construction centers the auxiliary circle at \(O\) with radius \(OP/2\). Identify the first invalid step, state the corrected center and radius construction, and explain how diameter \(OP\) creates right angles at the intersection points.
Audit this construction by its invariant: the auxiliary circle must actually have the center-to-external-point segment as a diameter. We’ll test the proposed center and radius against endpoint incidence, correct the first failed step, and verify that the correction restores the right angles needed for tangency.
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