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All courses Geometry · G-C.4 17 of 57
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 \(P\).
Unworked tangent diagram showing only the supplied configuration and givens. Open full size
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Hint

Look at point A first. Since A lies on the auxiliary circle whose diameter is OP, what kind of angle is ∠OAP?

Use the angle-in-a-semicircle idea: any angle that subtends diameter OP is 90°. Then use the tangent fact that a line perpendicular to a radius at the point on the circle is tangent there.

Solution walkthrough

01

Read the two-circle construction

\[\begin{aligned} OP~\text{is}~a~\text{diameter}~\text{of}~\text{the}~\text{auxiliary}~\text{circle} \\ A~\text{and}~B~\text{lie}~\text{on}~\text{both}~\text{circles} \end{aligned}\]

The original circle has center O. The auxiliary circle uses OP as a diameter and intersects the original circle at A and B, exactly as shown in the inspected source.

02

Create right angles from the diameter

\[\begin{aligned} \angle~OAP=90^{\circ} \\ \angle~OBP=90^{\circ} \end{aligned}\]

An angle inscribed in a semicircle is a right angle. Both angles subtend the auxiliary circle's diameter OP, so OA is perpendicular to PA and OB is perpendicular to PB.

03

Apply the radius-tangent criterion

\[\begin{aligned} OA~\text{is}~a~\text{radius}~\text{and}~OA~\text{perpendicular}~PA~->~PA~\text{tangent}~\text{at}~A \\ OB~\text{is}~a~\text{radius}~\text{and}~OB~\text{perpendicular}~PB~->~PB~\text{tangent}~\text{at}~B \end{aligned}\]

A line perpendicular to a radius at the radius endpoint is tangent to the original circle. Thus the segments from P touch the circle at the shared intersection points.

04

Name the requested points

\[\text{tangent}~\text{points}:~A~\text{and}~B\]

The construction produces two tangent segments, PA and PB, so their points of tangency are A and B.

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Another way

  1. Use the right-angle marks in the completed construction directly with the radius-tangent converse theorem.

!

Common mistake

Do not name O or P as a tangent point. A tangent point must lie on the original circle and be the endpoint where a tangent is perpendicular to a radius; those points are A and B.

Completed tangent answer diagram with exact construction or verification.
tangent points: A and B

Solution walkthrough video