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.

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

$\Box$

Title	some proofs for triangle theorems
Canonical name	SomeProofsForTriangleTheorems
Date of creation	2013-03-22 14:03:55
Last modified on	2013-03-22 14:03:55
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 51-00