如图,△ABC中,∠ABC=2∠C,BD平分∠ABC. 求证:AB•BC=AC•CD.

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如图,△ABC中,∠ABC=2∠C,BD平分∠ABC. 求证:AB•BC=AC•CD.
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如图,△ABC中,∠ABC=2∠C,BD平分∠ABC. 求证:AB•BC=AC•CD.
如图,△ABC中,∠ABC=2∠C,BD平分∠ABC. 求证:AB•BC=AC•CD.

如图,△ABC中,∠ABC=2∠C,BD平分∠ABC. 求证:AB•BC=AC•CD.

证明:∵BD平分∠ABC
      ∴∠ABD=∠CBD=∠ABC/2
      又∠ABC=2∠C
      从而∠ABD=∠C
     又∠A是△ABC相与△ADB的公共角
     ∴△ABC∽△ADB
     从而AB/AC=AD/AB ①
     ∵BD平分∠ABC
     ∴AD/AB=CD/BC ②
     由①②得AB/AC=CD/BC
     ∴AB•BC=AC•CD