Question

已知等边三角形ABC中,点P是三角形ABC内任意一点,设点P到三角形ABC三边AB、BC、AC的距离为h1、h2、h3,

Answer 1

设a为正△ABC边长;

(1)当P为△ABC内一点时,连接P与各顶点,得△PAB,△PAC,△PBC.此3个△的面积和等于△ABC的面积;

而△PAB=1/2*a*h1,△PAC=1/2*a*h2,△PBC=1/2*a*h3,△ABC=1/2*a*h,

又因S△PAB+S△PAC+S△PBC=S△ABC,即

1/2*a*h1+1/2*a*h2+1/2*a*h3=1/2*a*h;

化简,得:h1+h2+h3=h.