Question

已知DEF分别是锐角△ABC的三边BC,CA,AB上的点,AD,BE,CF,交于P,AP=BP=CP=a,PD=x,PE

Answer 1

如图,作PM⊥BC于M,AN⊥BC于N

S△PBC=1/2PM*BC

S△ABC=1/2AN*BC

S△PBC/S△ABC=PM/AN=PD/AD=x/(x+6)

同理S△PAC/S△ABC=y/(y+6),S△PAB/S△ABC=z/(z+6)

S△ABC=S△PBC+S△PAC+S△PAB

∴x/(x+6)=y/(y+6)=z/(z+6)=1

即1-6/(x+6)+1-6(y+6)+1-6/(z+6)=1

∴3/(x+6)+3/(y+6)+3/(z+6)=1

3（yz+zx+xy）+36（x+y+z）+324

=xyz+6（xy+yz+zx）+36（x+y+z）+216

∵xy+yx+zx=28

∴xyz=108-3（xy+yz+zx）=24