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Theorems for Segments and Circles

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Theorems for Segments and Circles

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Theorems for Segments and Circles

Theorems for Segments and Circles

Theorems for Segments and Circles

Theorems for Segments and Circles

Figure %: Chords of the same circle that intersect

In the figure above, chords QR and ST intersect. The theorem states that the product of QB and BR is equal to the product of SB and BT.

Theorem 2

Every secant segment is divided into two segments by the circle it intersects. The internal segment is a chord, and the external segment is the segment with one endpoint at the intersection of the secant segment and the circle, and the other endpoint at the fixed point outside the circle. Given these conditions, a theorem states that when two secant segments share an endpoint not on the circle, the products of the lengths of each secant segment and its external segment are equal.

Figure %: Secant segments that share an endpoint not on the circle
In the figure above, the secant segments DE and FE share an endpoint, E, outside the circle. The theorem states that the product of the lengths of DE and ME is equal to the product of the lengths of FE and NE.

Theorem 3

A similar theorem exists when a secant segment and a tangent segment share an endpoint not on the circle. This theorem states that the length of the tangent segment squared is equal to the product of the secant segment and its external segment.

Figure %: A secant segment and a tangent segment that share an endpoint not on the circle
In the figure above, secant segment QR and tangent segment SR share an endpoint, R, not on the circle. The theorem states that the length of SR squared is equal to the product of the lengths of QR and KR.

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