Question

如图,三角形ABC的周长为l,面积为s,其内切圆的圆心为O,半径为r,求r=2S/l.

Answer 1

设圆O与AB,AC,BC切于D,E,F,连OD,OE,OF

∴OD=OE=OF=r

∵S△ABC=S△OAB+S△OAC+S△OBC

∴S=(1/2)AB×OD+(1/2)AC×OE+(1/2)BC×OF=(1/2)AB×r+(1/2)AC×r+(1/2)BC×r=(1/2)(AB+AC+BC)×r

∴r=2S/(AB+AC+BC)=2S/l