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# What is the tangent secant theorem?

**The tangent-secant theorem states that if two secant segments share an endpoint outside of a circle, the product of one segment length and the length of its external segment equals the product of the other segment's length and the length of its external segment.** It is a special subtype of another mathematical theorem: the power of a point theorem.

The other specialized forms of the power of a point theorem are the chord-chord theorem and the secant-secant theorem. The power of a point theorem and its subtypes date back to ancient Greece, with their first confirmed appearance in Euclid's mathematical textbook "Elements" in 300 BC. Euclid's work with them is based on earlier proofs by Hippocrates of Chios, making the basis of the theorem nearly a century older than Euclid's work. Jakob Steiner, a German mathematician, further refined the theorem in 1826.

A common practical use of the tangent-secant theorem is in putting together sports plays. For example, it can be used to determine the optimal path in which to kick a soccer goal. It is also traditionally useful in navigation, where it is used to calculate the distance to the horizon, though this use has become less valuable with the invention of modern navigational equipment.

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