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射影定理 (又叫欧几里德(Euclid)定理)

直角三角形射影定理

直角三角形射影定理(又叫欧几里德(Euclid)定理):直角三角形中,斜边上的高是两直角边在斜边上射影的比例中项。每一条直角边是这条直角边在斜边上的射影和斜边的比例中项。公式Rt△ABC中,∠BAC=90°,AD是斜边BC上的高,则有射影定理如下: \[ \begin{array}{*{20}c} {1.} & {AD^2 = BD \cdot DC} \\ {2.} & {AB^2 = BD \cdot BC} \\ {3.} & {AC^2 = CD \cdot BC} \\ {4.} & {AB \cdot AC = BC \cdot AD} \\ \end{array} \]

直角三角形

直角三角形射影定理的证明

在△BAD与△BCD中,∵∠BDA=∠BDC=90°,且∠DBC+∠C=90°, ∴∠A=∠C,又∵∠BDA=∠BDC=90,∴△BAD∽△CBD,∴ AD/BD=BD/CD,即BD2=AD·DC.其余同理可得可证.

任意三角形射影定理

任意三角形射影定理又称“第一余弦定理”:△ABC的三边是abc,它们所对的角分别是ABC,则有 \[ \begin{array}{l} a = b \cdot \cos C + c \cdot \cos B \\ b = c \cdot \cos A + a \cdot \cos C \\ c = a \cdot \cos B + b \cdot \cos A \\ \end{array} \] 由三角形的边角关系可以容易证明该定理

任意三角形射影定理与余弦定理的关系

将任意三角形射影定理的三条公式作如下处理: \[ \left\{ \begin{array}{l} a^2 = ab \cdot \cos C + ac \cdot \cos B \\ b^2 = bc \cdot \cos A + ab \cdot \cos C \\ c^2 = ac \cdot \cos B + bc \cdot \cos A \\ \end{array} \right. \] 从上述三条等式中任取两条相加然后减去第三条等式: \[ \left\{ \begin{array}{l} a^2 + b^2 - c^2 = 2ab \cdot \cos C \\ b^2 + c^2 - a^2 = 2bc \cdot \cos A \\ c^2 + a^2 - b^2 = 2ac \cdot \cos B \\ \end{array} \right. \] 进一步处理,即可得到任意三角形余弦定理。

面积射影定理

平面图形射影面积等于被射影图形的面积S乘以该图形所在平面与射影面所夹角的余弦。

cosθ=S射影/S(平面多边形及其射影的面积分别是S原,S射影,它们所在平面所成锐二面角的为θ。)



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参阅
  1. 数学 - 数学符号 - 数学索引
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  3. 初等数学 = 小学数学 + 中学数学 ( 初中数学 + 高中数学 )
  4. 高等数学 = 基础数学 ( 代数 + 几何 + 分析 ) + 应用数学
  5. 公式 - 定理 - - 函数图 - 曲线图 - 平面图 - 立体图 - 动画 - 画画
  6. 书单 = 数学 + 物理 + 化学 + 计算 + 医学 + 英语 + 教材 - QQ群下载书
  7. 数学手册计算器 = 数学 + 手册 + 计算器 + 计算机代数系统
  8. 检测 - 例题 :

`(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)/x*y` == ? `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 = y*sin(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(4x)*y-exp(4x)=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) - x*y-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/(x-y^2-1)` == ? `y^((2))(x)*(x^2-2x*y+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^((sin(x)))(x)=2y-exp(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-x*y=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^2-2x*y=0` == ? `y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0` == ? `d^sin(x)/dx^sin(x) y -2y-exp(x)=0` == ? `d^cos(x)/dx^cos(x) y -4y-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/x*sin(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^3*x^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^0.5-2x*y-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)*(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^2*y^2-8x^2=0` == ? `(d^0.5y)/dx^0.5-2x*y^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` == ? `2*x/y-3*y^2/x^4+(-x^2/y^2+1/y^(1/2)+2*y/x^3)*dy/dx = 0` == ?


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