Question

已知三角形两边几角平分线对应相等,能否证明这两个三角形全等?

Answer 1

证：过C作CP//AB交AD延长线于P,过C'作C'P'//A'B'交A'D'延长线于P'

∴∠APC=∠BAP=∠CAP

∴AC=PC

同理,A'C'=P'C'

又∵AC=A'C'

∴PC=AC=A'C'=P'C'

∵CP//AB

∴△ABD∽△PCD

∴AD/PD=AB/PC

∴AD/PD=AB/AC

同理,A'D'/P'D'=A'B'/A'C'

∴AD/PD=A'B'/A'C'=AB/AC=A'D'/P'D'

又∵AD=A'D'

∴PD=P'D'

∴AP=A'P'

又∵AC=A'C',PC=P'C'

∴△ACP≌△A'C'P'

∴∠PAC=∠P'A'C'

又∵∠PAC=∠BAD,∠P'A'C'=∠B'A'D'

∴∠PAC=∠BAD=∠P'A'C'=∠B'A'D'

∴∠BAC=∠B'A'C'

又∵AB=A'B',AC=A'C'

∴△ABC≌△A'B'C'