卡诺重心定理以及莱布尼兹公式卡诺重心定理：G为三角形ABC的重心,P为三角形ABC所在平面上任意一点.求证：PA^2+P

GA^2 + PG^2 = PA^2 + 2GA*PGcos(AGP)
GB^2 + PG^2 = PB^2 + 2GB*PGcos(BGP)
GC^2 + PG^2 = PC^2 + 2GC*PGcos(CGP)
GA^2 + GB^2 + GC^2 + 3PG^2 = PA^2 + PB^2 + PC^2 + 2PG[GA*cos(AGP) + GB*cos(BGP) + GC*cos(CGP)]
延长射线AG,交BC于D,继续延长,使得GD = DE = AG/2.
连接EB,EC,
四边形GBEC为平行四边形.
EB = GC
延长射线PG,
过点B作PG的延长线的垂线,垂足为F.
过点E作PG的延长线的垂线,垂足为H.
BE与PG的延长线的交点为点Q.
则,因GC//BE,角CGP = 角EQG = 角BQF
GH = GE*cos(EGH) = GA*cos(AGP)
HF = EB*cos(BQF) = GC*cos(EQG) = GC*cos(CGP)
而
GH + HF = GF = GB*cos(BGF) = GB*cos(PI-BGP) = -GB*cos(BGP),
因此,
GA*cos(AGP) + GB*cos(BGP) + GC*cos(CGP) = 0,
GA^2 + GB^2 + GC^2 + 3PG^2
= PA^2 + PB^2 + PC^2 + 2PG[GA*cos(AGP) + GB*cos(BGP) + GC*cos(CGP)]
= PA^2 + PB^2 + PC^2
利用上面的结论,
令P与A重合,有
GA^2 + GB^2 + GC^2 + 3GA^2
= AB^2 + AC^2 ...(1)
令P与B重合,有
GA^2 + GB^2 + GC^2 + 3GB^2
= AB^2 + BC^2 ...(2)
令P与C重合,有
GA^2 + GB^2 + GC^2 + 3GC^2
= BC^2 + AC^2 ...(3)
(1),(2),(3)相加,有
3[GA^2 + GB^2 + GC^2] + 3[GA^2 + GB^2 + GC^2] = 2[AB^2 + BC^2 + AC^2],
GA^2 + GB^2 + GC^2 = [AB^2 + BC^2 + AC^2]/3 = (a^2 + b^2 + c^2)/3.
证毕.

PA=PG+GA
PB=PG+GB
PC=PG+GC
PA^2+PB^2+PC^2
=PA.PA+PB.PB+PC.PC
=(PG+GA).(PG+GA)+(PG+GB).(PG+GB)+(PG+GC).(PG+GC)
=GA2+GB^2+GC^2+3PG^2+2PG(GA+GB+GC)
=GA2+GB^2+GC^2+3PG^2
=...