Question

已知,如图AB=AC,AD=AE,∠ BAC=∠ DAE=90° ,M是BE中点,求证：AM⊥DC

Answer 1

AM于CD的交点为点N,延长AM到F,使MF=AM

∵BM=EM

∴ABFE是平行四边形

∴BF=AE ∠ABF+∠BAE=180°

∵∠BAC=∠DAE=90°

∴∠CAD+∠BAE=180°

∴∠ABF=∠CAD

∵BF=AE AD=AE

∴BF=AD

∵AB=AC

∴△ABF ∽△CAD

∴∠BAF=∠ACD

∵∠BAC=90°

∴∠BAF+∠CAN=90°

∴∠ACD+∠CAN=90°

∴∠ANC=90°

∴AM⊥CD