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Mathematics LibreTexts

9.1: Angle between a tangent line and a chord

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Theorem 9.1.1

Let Γ be a circle with the center O. Assume the line (XQ) is tangent to Γ at X and [XY] is a chord of Γ. Then

2QXYXOY.

Equivalently,

QXY12XOY or QXY12XOY+π.

Proof

Note that XOY is isosceles. Therefore, YXO=OYX.

Applying Theorem 7.4.1 to XOY, we get

截屏2021-02-18 上午11.12.03.png

πYXO+OYX+XOY2YXO+XOY.

By Lemma 5.6.2, (OX)(XQ), Therefore,

QXY+YXO±π2.

Therefore,

2QXYπ2YXOXOY.


This page titled 9.1: Angle between a tangent line and a chord is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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