Question 160
In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB, BC, CD, DA$, respectively. Prove that
$$
PA \cdot AB + RC \cdot CD = \frac{1}{2}\left(AD^2 + BC^2\right)
$$
if and only if
$$
QB \cdot BC + SD \cdot DA = \frac{1}{2}\left(AB^2 + CD^2\right).
$$
$$
PA \cdot AB + RC \cdot CD = \frac{1}{2}\left(AD^2 + BC^2\right)
$$
if and only if
$$
QB \cdot BC + SD \cdot DA = \frac{1}{2}\left(AB^2 + CD^2\right).
$$