# What is the geometric meaning of this vector equality? $\vec{BC}\cdot\vec{AD}+\vec{CA}\cdot\vec{BD}+\vec{AB}\cdot\vec{CD}=0$

I was doing some exercises for linear algebra. One of them was to prove that for any four points $$A, B, C, D \in \mathbb{R}^3$$ the following equality holds:
$$\overrightarrow{BC} \cdot \overrightarrow{AD}\ +\ \overrightarrow{CA} \cdot \overrightarrow{BD}\ +\ \overrightarrow{AB} \cdot \overrightarrow{CD}\ = 0$$
The proof is easy; you just make three vectors starting in $$A$$ and then see that all the terms cancel out.

My question is: what is the geometric interpretation of this equality? How can I visualize it or understand its deeper meaning? Does this equality have a name or where can I read more about it?

I’m asking this because it turns out that it is not just a random equality and is rather useful. For example, if we want to prove the existence of orthocenter, we can do it surprisingly easily and quickly using this equality.

Let $$O$$ be the orthocenter $$O$$ of $$\triangle ABC$$. Then
\begin{align} &\overrightarrow{AB}\cdot\overrightarrow{CD} \ +\ \overrightarrow{BC}\cdot\overrightarrow{AD} \ +\ \overrightarrow{CA}\cdot\overrightarrow{BD}\\ =\ &\left(\overrightarrow{AB}\cdot\overrightarrow{CO} \ +\ \overrightarrow{BC}\cdot\overrightarrow{AO} \ +\ \overrightarrow{CA}\cdot\overrightarrow{BO}\right) + \left(\overrightarrow{AB}\cdot\overrightarrow{OD} \ +\ \overrightarrow{BC}\cdot\overrightarrow{OD} \ +\ \overrightarrow{CA}\cdot\overrightarrow{OD}\right)\\ =\ &\left(\overrightarrow{AB}\cdot\overrightarrow{CO} \ +\ \overrightarrow{BC}\cdot\overrightarrow{AO} \ +\ \overrightarrow{CA}\cdot\overrightarrow{BO}\right) + \left(\overrightarrow{AB}\ +\ \overrightarrow{BC}\ +\ \overrightarrow{CA}\right)\cdot\overrightarrow{OD}\tag{\dagger}\\ =\ &0+0=0.\\ \end{align}
The first bracket on line $$(\dagger)$$ is zero because every side of $$\triangle ABC$$ is perpendicular to the altitude dropped from the opposite vertex. The second bracket is zero because it is the sum of directed edges of a closed circuit.
In short, the identity is basically a cyclic sum of expressions of the form “side dot altitude” on $$\mathbb R^2$$, but another cyclic sum of the form “side dot $$\overrightarrow{OD}$$” has been added to conceal the significance of the orthocenter and make the identity present in $$\mathbb R^3$$.