Question

△ABC中,D为BC上一点,且AB=AD,求证AC²-AB²=BC·DC

Answer 1

证明,在AC上找一点E,使AE=AD,延长CA至F,使AF=AE,连接BF

∵△AFB△ABD△ADE都是等腰三角形

∴2∠AED﹢2∠ABD﹢2∠FBA=540-180

∴∠ADE﹢∠ABD﹢∠FBA=180

∴∠ABD﹢∠FBA=180-∠AED=∠DEC,即∠FBD=∠DEC

∴△FBC∽△DEC

∴FC﹕BC=DC﹕EC

FC=AC﹢AD=AC﹢AB,EC=AC‐AE=AC‐AB

∴DC*BC=(AC﹢AB)(AC‐AB)=AC²‐AB²