Question

在三角形ABC中,∠B=2∠C,AD是∠BAC的平分线.试证明:AC=AB+BD解法.

Answer 1

1.延长AB至E,使AE=AC,则△ABE≌△ADC

∴∠E=∠C,又∠ABD=∠E+∠EDB=2∠C=2∠E,∴∠EDB=∠E ∴BE=BD

∴AC=AE=AB+BE=AB+BD

2.延长DB至E,使BE=AB,则∠E=∠BAE

又∠ABD=∠E+∠BAE,∴∠ABD=2∠BAE=2∠C,∴∠BAE=∠C,∴AC=AE

又∠ADE=∠C+∠DAC,∠DAE=∠BAE+∠DAB,∴∠ADE=∠DAE ∴AE=ED

∴AC=ED=EB+BD=AB+BD