We have been asked to prove the Segments of Secants and Tangents Theorem for the special case when the secant segment contains the center of the circle. To prove this theorem, we will write a two-column proof. Let's first recall the theorem!

Segments of Secants and Tangents Theorem
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

By this theorem we can draw a diagram that models the special case.

Note that two-column proof lists each statement on the left and the justification on the right. Each statement must follow logically from the steps before it. In a two-column proof, we always start by stating the given information.

Statement 1) Reason 1) \overline{E A} is a tangent segment . \overline{E D} is a secant segment that contains the center of ⊙ O . Given

To continue our proof, as it was suggested we will use the Tangent Line to Circle Theorem. By this theorem we know that

\overline{O A}

is perpendicular to

\overline{E A}

at point

A .

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