已知:如图,同心圆O,大圆的弦AB=CD,且AB是小圆的切线,切点为E.求证:CD是小圆的切线.

问题描述:

已知:如图,同心圆O,大圆的弦AB=CD,且AB是小圆的切线,切点为E.求证:CD是小圆的切线.

证明:如右图所示,连结OE,过O作OF⊥CD于F.
∵AB与小⊙O切于点E,
∴OE⊥AB,
∵AB=CD,
∴OE=OF(同圆等弦的弦心距相等),
∴CD与小⊙O相切.