Tangent-Secant Theorem: Proving Equal Length Segments in Circles

If two tangent lines share a endpoint (or point of intersection) outside of a circle creating segments with the other endpoint being the points of tangency, then those segments are…

Congruent

equal in length.

This is a property of tangent lines to circles. It is sometimes called the Tangent-Secant Theorem.

Suppose that we have a circle with center O, and two tangent lines AB and CD that intersect at a point P outside the circle. Let E and F be the points of tangency on the circle for AB and CD, respectively.

Using the properties of tangent lines, we can see that angles PAE and PBF are both right angles. Therefore, triangles PAE and PBF are both right triangles. Furthermore, they share the hypotenuse PF (which is equal to PE since they are radii of the same circle). Using the Pythagorean theorem on each triangle, we can see that:

PA^2 + AE^2 = PE^2

and

PB^2 + BF^2 = PF^2

But since PE = PF, we can combine these equations:

PA^2 + AE^2 = PB^2 + BF^2

This means that the segments AE and BF have the same length. Therefore, if two tangent lines share an endpoint outside the circle, the segments they create are equal in length.

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