Question

已知：如图，M为正方形ABCD的边AB上一点，BP垂直于CM于点P，N是BC上一点，PD垂直于PN。求证BM=BN

已知：如图，M为正方形ABCD的边AB上一点，BP垂直于CM于点P，N是BC上一点，PD垂直于PN。求证BM=BN

Answer 1

∠BPL=∠NPD=90度
=> ∠BPN=∠CPD
=> ∠PBN+∠PCB=∠PCD+∠PCB=90度
=> PBN=∠PCD
=> △PBN相似于△PCD
=> PB:PC=BN:CD

PB:PC=tan ∠PCB=BM:BC
=> BN:CD=BM:BC
CD=BC => BN=BM

Answer 2

∠BPC=∠NPD=90度
=> ∠BPN=∠CPD
=> ∠PBN+∠PCB=∠PCD+∠PCB=90度
=> PBN=∠PCD
=> △PBN∽△PCD
=> PB:PC=BN:CD

PB:PC=tan ∠PCB=BM:BC
=> BN:CD=BM:BC
CD=BC => BN=BM