作业帮求证一道关于内心的几何题目!
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/17 01:59:36
求证一道关于内心的几何题目!
作△ABC的内切⊙I,过点I分别作AC、AB的垂线IE、IF,E、F为垂足,
则有ID+IF=IE=AE=AF,由切线长定理可得:BF=BD,CE=CD,
∵S△ABC=AB×AC/2=(BF+ID)×(CF+AE)/2=(CD+ID)×(BD+ID)/2
=[CD×BD+ID(BD+CD)+(ID^2)]/2=CD×BD/2+(ID×BC+(ID^2))/2,
∵ID×BC/2=S△ICB,(ID^2)/2=S△AEI,S△ICB+S△AEI=S△ABC/2,
∴(ID×BC+(ID^2))/2=CD×BD/2=S△ABC/2,
∴S△ABC=BD×CD.
则有ID+IF=IE=AE=AF,由切线长定理可得:BF=BD,CE=CD,
∵S△ABC=AB×AC/2=(BF+ID)×(CF+AE)/2=(CD+ID)×(BD+ID)/2
=[CD×BD+ID(BD+CD)+(ID^2)]/2=CD×BD/2+(ID×BC+(ID^2))/2,
∵ID×BC/2=S△ICB,(ID^2)/2=S△AEI,S△ICB+S△AEI=S△ABC/2,
∴(ID×BC+(ID^2))/2=CD×BD/2=S△ABC/2,
∴S△ABC=BD×CD.