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Homesome proofs for triangle theorems

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# some proofs for triangle theorems

In the following, only Euclidean geometry is considered.

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

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The area formula $A=rs$ 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.

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

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