卡诺重心定理以及莱布尼兹公式

卡诺重心定理以及莱布尼兹公式

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。