Power of a point

Japanese school year: Math A

What you learn

Understand the Power of a Point theorem, relating line segments formed by intersecting chords, secants, and tangents of circles. This topic relies on prior knowledge of triangle similarity and inscribed angle properties. It is a powerful tool for calculating unknown lengths in circle geometry and solving advanced proofs.

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Key points

Known as the power of a point theorem, this applies when two lines through point $P$ intersect a circle at $A, B$ and $C, D$ . The products $PA \cdot PB$ and $PC \cdot PD$ are always equal.

PA \cdot PB = PC \cdot PD

This states the power of a point theorem for a secant and tangent from external point $P$ . For circle intersections $A, B$ and tangency point $T$ , $PA \cdot PB$ equals $PT^2$ .

PA \cdot PB = PT^2

This is the converse of the power of a point theorem. When two lines $AB$ and $CD$ intersect at $P$ such that $PA \cdot PB = PC \cdot PD$ , the four points $A, B, C, D$ lie on one circle.

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Practice 1Coming soon