Application of Dot and Cross Products in Geometry

Placing a convenient vertex at the origin turns several classical Euclidean-geometry theorems into short vector computations.

Apollonius's theorem (Example 6.6). If $D$ is the midpoint of side $BC$ of $\triangle ABC$, taking $A$ as origin with $\vec{AB}=\vec b,\ \vec{AC}=\vec c$ gives $\vec{AD}=\tfrac{\vec b+\vec c}{2}$ and $\vec{BD}=\tfrac{\vec c-\vec b}{2}$; computing $|AD|^2+|BD|^2$ and simplifying (the cross terms cancel) gives

Concurrency of altitudes (Example 6.7). With the orthocentre $O$ as origin, $\vec{OA}=\vec a,\vec{OB}=\vec b,\vec{OC}=\vec c$. Since altitude $AD\perp BC$ means $\vec a\cdot(\vec c-\vec b)=0$, and altitude $BE\perp CA$ means $\vec b\cdot(\vec a-\vec c)=0$; adding these two equations gives $\vec c\cdot(\vec a-\vec b)=0$, i.e. $\vec{OC}\perp\vec{BA}$ — so the perpendicular from $C$ to $AB$ also passes through $O$. All three altitudes meet at one point.

Medial-triangle area (Example 6.8). If $D,E,F$ are the midpoints of $BC,CA,AB$ of $\triangle ABC$ (origin at $A$), a short computation with $\vec{DE},\vec{DF}$ shows $\text{area}(\triangle DEF)=\tfrac14\,\text{area}(\triangle ABC)$ — the medial triangle always has a quarter of the original's area. …