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AM是△ABC的BC边上的中线，则有

$AB^2 + AC^2 = 2\left( {AM^2 + BM^2 } \right)$

中线定理证明

过点A作BC的垂线AM，垂足为D．在Rt△ABD中，由勾股定理可知，

$AB^2 = AD^2 + BD^2$ ；同理，

$AC^2 = AD^2 + CD^2$ ，

$AM^2 = AD^2 + DM^2$

故

$AB^2 + AC^2 = 2AD^2 + BD^2 + CD^2$

$= 2\left( {AM^2 - DM^2 } \right) + \left( {BM + DM} \right)^2 + \left( {CM - DM} \right)^2$

$= 2\left( {AM^2 - DM^2 } \right) + BM^2 + DM^2 + CM^2 + DM^2$

斯德瓦尔特定理

△ABC中，D是BC上任意一点，则有

$AB^2 \cdot CD + AC^2 \cdot BD = AD^2 \cdot BC + BD \cdot CD \cdot BC$

斯德瓦尔特定理证明

过A作BC的垂线，垂足为E，则有

$AB^2 = AE^2 + BE^2$ ，

$AC^2 = AE^2 + CE^2$ ，

$AD^2 = AE^2 + DE^2$ ，故

$AB^2 \cdot CD + AC^2 \cdot BD - AD^2 \cdot BC$

$= \left( {AE^2 + BE^2 } \right) \cdot CD + \left( {AE^2 + CE^2 } \right) \cdot BD - \left( {AE^2 + DE^2 } \right) \cdot BC$

$= AE^2 \left( {CD + BD - BC} \right) + BE^2 \cdot CD + CE^2 \cdot BD - DE^2 \cdot BC$

$= BE^2 \cdot CD + CE^2 \cdot BD - DE^2 \cdot BC = \left( {BD + DE} \right)^2 \cdot CD + \left( {CD - DE} \right)^2 \cdot BD$

$= BE^2 \cdot CD + CE^2 \cdot BD - DE^2 \cdot BC$

$= \left( {BD + DE} \right)^2 \cdot CD + \left( {CD - DE} \right)^2 \cdot BD$

$- DE^2 \cdot BC = \left( {BD^2 + DE^2 + 2BD \cdot DE} \right) \cdot CD + \left( {CD^2 + DE^2 - 2CD \cdot DE} \right) \cdot BD$

$- DE^2 \cdot BC = BD^2 \cdot CD + CD^2 \cdot BD + DE^2 \cdot BC - DE^2 \cdot BC$

$= BD^2 \cdot CD + CD^2 \cdot BD$

由斯德瓦尔特定理可以得出很多有用的结论，比如上例，令本例中

$BD = CD$ ，则很快得出上例的结论以及中线长的公式，一般地，只要

$\Delta ABC$ 的三条边已知，BC上一点D的位置已知，则AD的长度便可直接求出来．此外，用余弦定理证明斯德瓦尔特定理也是很快的：

在△ABC中，由余弦定理可知，

$AB^2 = AD^2 + BD^2 - 2AD \cdot BD\cos \angle ADB$ ；

在△ACD中，由余弦定理可知，

$AC^2 = AD^2 + CD^2 - 2AD \cdot CD\cos \angle ADC$ ;

故

$AB^2 \cdot CD + AC^2 \cdot BD - AD^2 \cdot BC$

$= AD^2 \cdot CD + BD^2 \cdot CD + AD^2 \cdot BD + CD^2 \cdot BD$

$- AD^2 \cdot BC = AD^2 \cdot BC + BD \cdot CD \cdot BC - AD^2 \cdot BC$

$= BD \cdot CD \cdot BC$

问题？

$(d^0.5y)/dx^0.5 = sin(x-1)sin(y-1)$ == ? $(d^0.5y)/dx^0.5 -cosh(y)-sinh(y)=0$ == ? $(d^1.6y)/(dx^1.6)-int y(x) (dx)^(0.8)-y-exp(x)=0$ == ? $int y(x) (dx)^0.5 -y-exp(x)$ =0 == ? $(d^0.5y)/dx^0.5-exp(y)x=0$ == ? $(d^0.5y)/dx^0.5-exp(y)y=0$ == ? $(d^0.5y)/dx^0.5=cos(x)/xy$ == ? $y(dy^0.5)/dx^0.5-sqrt(x)-1=0$ == ? $(d^1.2y)/(dx^1.2)-2(d^0.6y)/dx^0.6+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5=cos(y)exp(x)x$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5-exp(y)sqrt(x)=0$ == ? $(d^1.6y)/(dx^1.6)-3 (d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5$ +log(y-1)-exp(x)-x=0 == ? $(d^0.5y)/dx^0.5-exp(y)sin(x)=0$ == ? $(d^0.5y)/dx^0.5 = ysin(x)/x$ == ? $y^((0.5))(x) -4 exp(x)y-exp(x)=0$ == ? $(dy^0.5)/dx^0.5 = 1/(x-y)$ == ? $dy/dx-(d^0.5y)/dx^0.5$ - y - exp(x)=0 == ? $(dy)/dx -exp(y-1)-x-x^2=0$ == ? $(d^1.2y)/(dx^1.2)-3dy^0.6/dx^0.6+2y-exp(x)=0$ == ? $dy/dx-(d^0.5y)/dx^0.5-y-1$ =0 == ? $(d^0.5y)/dx^0.5-cos(y)sin(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $dy/dx-exp(y-1)-exp(x)=0$ == ? $(dy)/dx - 2(d^0.5y)/dx^0.5-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5 -e^(4x)-y$ =0 == ? $y^((0.5))(x) - exp(x)y-exp(x)=0$ == ? $y^((0.5))(x) - exp(x)y-4exp(x)=0$ == ? $(dy)/dx -3(d^0.5y)/dx^0.5 +2y-exp(x)=0$ == ? $y(d^0.5y)/dx^0.5-sqrt(x)-1=0$ == ? $y^((1))(x)-exp(y-1)-x=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-2y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $(d^0.5y)/dx^0.5 - log(y-1) - exp(x) + x=0$ == ? $(dy)/dx +asin(y-1) - cos(x)-x=0$ == ? $(d^1.6y)/(dx^1.6)-3(d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(dy)/(dx) -sqrt(y-1)-x-1 =0$ == ? $(dy)/(dx) -exp(y-1)-exp(x) = 0$ == ? $(dy)/dx$ +asinh(y-1)-cosh(x)-x =0 == ? $((d^(1/2)y)/dx^(1/2))^2 -3y (dy^0.5)/dx^0.5 + 2y^2 = 0$ == ? $(dy^0.5)/dx^0.5 = cos(x)cos(y-1)$ == ? $(d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0$ == ? $(dy^0.5)/dx^0.5 = sin(x-1)exp(y-1)$ == ? $y(d^2y)/dx^2-(dy/dx)^2+1=0$ == ? $y^((1))(x)-exp(y-1)-log(x)=0$ == ? $(d^2y)/dx^2 exp(x)- exp(y-1)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(dy)/dx -3 (d^0.5y)/dx^0.5+2y-exp(x)=0$ == ? $y^((0.5))(x) - xy-x=0$ == ? $y(dy^3)/dx^3-x^3-3x^2-3x-1=0$ == ? $y^((1.8))(x)-2y^((0.9))(x) +y-1=0$ == ? $y^((0.5))(x)=1/(xy-1)$ == ? $y^((2))(x)y^2-x^2-2x-1=0$ == ? $((d^0.5y)/dx^0.5)^2 -5(d^0.5y)/dx^0.5 +6=0$ == ? $y^((0.5))(x) -2 exp(x)y-4exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $y^(0.5)(x)=2yexp(x)$ == ? $y^((0.5))(x)-exp(x)y^2=0$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $y^((1))(x)-y^2-xy=0$ == ? $y^((1))(x)-y^((0.5))(x) -y-1=0$ == ? $y^((2))(x) -y^2-x^2=0$ == ? $y^((2))(x) -y^2-x^2-2xy=0$ == ? $y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0$ == ? $d^0.5/dx^0.5 y -2cos(y)exp(x)=0$ == ? $d^0.5/dx^0.5 y -4sin(y)exp(x)=0$ == ? $(d^0.5y)/dx^0.5=sin(x^2)y$ == ? $(d^0.5y)/dx^0.5-sin(x)sin(y)=0$ == ? $(d^0.5y)/dx^0.5-sinh(x)sinh(y)=0$ == ? $y^((1))(x)=exp(x-y)-x$ == ? $x(d^0.5y)/dx^0.5-y-2x=0$ == ? $(d^0.5y)/dx^0.5=sinh(x-1)sinh(y-1)$ == ? $y^((0.5))(x)-exp(-x)y^2=0$ == ? $(d^0.5y)/dx^0.5=y/xsin(x)$ == ? $(dy)/dx-sin(x-y)-1=0$ == ? $(d^2.5y)/dx^2.5=y(d^0.5y)/dx^0.5$ == ? $(d^0.5y)/dx^0.5=y(dy)/dx$ == ? $(d^(2-i)y)/dx^(2-i)- y+x=0$ == ? $(d^2y)/dx^2=y^3x^2$ == ? $y(d^2y)/dx^2-x^2-3x-1=0$ == ? $y(d^2y)/dx^2-2x^2-3x-1=0$ == ? $(y-x-1)(d^2y)/dx^2-3x-1=0$ == ? $y^2(d^2y)/dx^2-x^2-4x-4=0$ == ? $(y-x-1)(d^2y)/dx^2-x^2-4x-4=0$ == ? $y(d^2y)/dx^2-2x^2-2x-1=0$ == ? $y(d^3y)/dx^3-6x^3-3x^2-3x-1=0$ == ? $y^((0))(x)y^((1))(x)y^((2))(x)=x^2$ == ? $y^((3))(x)y^((2))(x)=y^((1/2))(x)$ == ? $y^((3))(x)=exp(x)y^((1))(x)y^((1/2))(x)$ == ? $y^((1/2))(x)y^((3))(x)=exp(x)$ == ? $y^((1/2))(x)y^((2))(x)=exp(x)$ == ? $(d^0.5y)/dx^0.5-2xy-1=0$ == ? $y^2(d^0.5y)/dx^0.5-x^2-4x-4=0$ == ? $exp(y-1)(d^0.5y)/dx^0.5-x=0$ == ? $y(d^2y)/dx^2-(x-2)(2x-4)=0$ == ? $y(d^3y)/dx^3-6x^3-4x^2-4x-1=0$ == ? $exp(y-1)(d^2y)/dx^2-exp(x)=0$ == ? $y^2(d^2y)/dx^2-x^2-1=0$ == ? $1/y^2(d^2y)/dx^2-x^2-1=0$ == ? $(y-x-1)(d^3y)/dx^3-(x-2)(2x-4)(3x-1)=0$ == ? $(d^0.5y)/dx^0.5-2x^2y^2-8x^2=0$ == ? $(d^0.5y)/dx^0.5-2xy^2-8x=0$ == ? $(d^0.5y)/dx^0.5-y^2-2y-2=0$ == ? $(d^0.5y)/dx^0.5-log(y-1)exp(x)=0$ == ? $y(d^2y)/dx^2-(dy/dx)^2-1=0$ == ? $(d^2y)/dx^2-asin(y-1)-sin(x)-x=0$ == ? $dy/dx(x--y)-x--y-1 = 0$ == ?

中线定理, 斯图尔特定理

中线定理

中线定理证明

斯德瓦尔特定理

斯德瓦尔特定理证明

问题 ？

问题？