some proofs for triangle theorems

In the following, only Euclidean geometry is considered.

The sum of three angles $A$ , $B$ , and $C$ of a triangle is $A+B+C=180^{\\circ}$ .

The following triangle shows how the angles can be found to make a half revolution, which equals $180^{\\circ}$ .

$\\Box$

The area $A=rs$ where $s$ is the semiperimeter $\\displaystyle s=\\frac{a+b+c}{2}$ and $r$ is the radius of the inscribed circle can be proven by creating the triangles $\\triangle BAO$ , $\\triangle BCO$ , and $\\triangle ACO$ from the original triangle $\\triangle ABC$ , where $O$ is the center of the inscribed circle.

$\\begin{array}[]{rl}A_{\\triangle ABC}&=A_{\\triangle ABO}+A_{\\triangle BCO}+A_{%\\triangle ACO}\\\\&\\\\&\\displaystyle=\\frac{rc}{2}+\\frac{ra}{2}+\\frac{rb}{2}\\\\&\\\\&\\displaystyle=\\frac{r(a+b+c)}{2}\\\\&\\\\&=rs\\end{array}$

$\\Box$

单词	SomeProofsForTriangleTheorems
释义	some proofs for triangle theorems In the following, only Euclidean geometry is considered. The sum of three angles $A$ , $B$ , and $C$ of a triangle is $A+B+C=180^{\\circ}$ . The following triangle shows how the angles can be found to make a half revolution, which equals $180^{\\circ}$ . $\\Box$ The area $A=rs$ where $s$ is the semiperimeter $\\displaystyle s=\\frac{a+b+c}{2}$ and $r$ is the radius of the inscribed circle can be proven by creating the triangles $\\triangle BAO$ , $\\triangle BCO$ , and $\\triangle ACO$ from the original triangle $\\triangle ABC$ , where $O$ is the center of the inscribed circle. $\\begin{array}[]{rl}A_{\\triangle ABC}&=A_{\\triangle ABO}+A_{\\triangle BCO}+A_{%\\triangle ACO}\\\\&\\\\&\\displaystyle=\\frac{rc}{2}+\\frac{ra}{2}+\\frac{rb}{2}\\\\&\\\\&\\displaystyle=\\frac{r(a+b+c)}{2}\\\\&\\\\&=rs\\end{array}$ $\\Box$
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