The following paragraph is extracted from original paper of Angle-Based Outlier Detection in High-dimensional Data [1].
Given a database $\mathcal{D} \subseteq \mathbb{R}$, a point $\vec{A} \in \mathcal{D}$, and a norm $||.|| : \mathbb{R}^d \rightarrow \mathbb{R_{0}^{+}}$. The scalar product is denoted by $<.,.>: \mathbb{R}^d \rightarrow \mathbb{R_{.}}$. For two points $\vec{B}$, $\vec{C} \in \mathcal{D}, \overline{BC}$ denotes the difference vector $ \vec{C}-\vec{B}$. The angle-based utlier factor $ABOF(\vec{A})$ is the variance over the angles between the difference vectors of $\vec{A}$ to all pairs of points in $\mathcal{D}$ weighted by the distance of the points:
$ABOF(\vec{A})$ = $VAR_{\vec{B},\vec{C} \in \mathcal{D}}$ ($\frac{<\overline{AB,AC}>}{||\overline{AB}||^2.||\overline{AC}||^2}$) = $\frac{\sum_{\vec{B} \in \mathcal{D}}\sum_{\vec{C} \in \mathcal{D}}(\frac{1}{||\overline{AB}||.||\overline{AC}||}.\frac{<\overline{AB,AC}>}{||\overline{AB}||^2.||\overline{AC}||^2})^2}{\sum_{\vec{B} \in \mathcal{D}}\sum_{\vec{C} \in \mathcal{D}}\frac{1}{||\overline{AB}||.||\overline{AC}||}}$ - $\Bigg( \frac{\sum_{\vec{B} \in \mathcal{D}}\sum_{\vec{C} \in \mathcal{D}}(\frac{1}{||\overline{AB}||.||\overline{AC}||}).\frac{<\overline{AB,AC}>}{||\overline{AB}||^2.||\overline{AC}||^2}}{\sum_{\vec{B} \in \mathcal{D}}\sum_{\vec{C} \in \mathcal{D}}\frac{1}{||\overline{AB}||.||\overline{AC}||}} \Bigg)^2$
What is the mathematical proof intuition so as to prove the equality of both sides of above equation. Does the author use the expansion formula for $VAR$ calculation on vectors?
[1]. Kriegel, Hans-Peter & Schubert, Matthias & Zimek, Arthur. (2008). Angle-based outlier detection in high-dimensional data. Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. 444-452. 10.1145/1401890.1401946.