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Let
- $P$ be an arbitrary reference point.
- $\triangle ABC$ be a triangle.
- $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$
Remark: $O$ is reserved for circumcenter.
| symbol | name | meaning |
|---|---|---|
| $G$ | centroid | center of gravity |
| $H$ | orthocenter | three “heights” are concurrent |
| $I$ | incenter | center of inscribed circle |
| $O$ | circumcenter | center of circumscribed circle |
Centroid
Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints.
Orthocenter
Let
- $H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$
- $\vec{h} = \overrightarrow{PH}$
\begin{align}
(\vec{h} - \vec{a}) \cdot (\vec{b} - \vec{c}) &= 0 \\
(\vec{h} - \vec{b}) \cdot (\vec{c} - \vec{a}) &= 0
\end{align}
Add these two equations together.
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