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More Theorems

For the entirety of this page, let:

• $r$ denote inradius
• $R$ denote circumradius
• $s$ denote semiperimeter i.e. $s=\dfrac{a_1+a_2+\dots+a_n}{2}$ for a $n$-gon.
• uppercase letter ($A,B,C\dots$) denote the angle at that vertex
• lowercase letter denote side length (if in triangle, the opposite side of the vertex with uppercase label)

You might find it fun to prove some of these yourself. Anyway, we have:

1. For any triangle, $\sin \dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{bc}}$ and $\cos \dfrac{A}{2} = \sqrt{\dfrac{(s)(s-a)}{bc}}$.
2. For any triangle $\cos A+\cos B+\cos C=1+\dfrac{r}{R}$.
3. For any triangle $\sin \dfrac{A}{2}\sin \dfrac{B}{2}\sin \dfrac{C}{2}=\dfrac{r}{4R}$.