Problem

A proposed construction uses the \(OP\)-diameter auxiliary circle and claims \(\text{two}~\text{real}~\text{tangents}\), but \(OP~\text{is}~\text{less}~\text{than}~\text{the}~\text{original}~\text{circle}'s~\text{radius}\). Identify the first failed condition, classify \(P\) relative to the circle, and state the correct real-tangent count.

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The point-location condition fails. Since OP is less than the radius, P is inside the circle and no real tangent from P exists.

The diameter is too short. Extend OP until it equals the original radius, then construct two tangents from the same point P.

P must lie on the circle. Move P to the circumference and use the two auxiliary-circle intersections as tangent points.

The auxiliary radius is too small. Double it while keeping interior point P, then use the two resulting intersections.

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