Question 23
Let $ABCD$ be a convex quadrilateral. The orthogonal projections of $D$ on the lines $BC$ and $BA$ are denoted by $A_1$ and $C_1$, respectively.
The segment $A_1C_1$ meets the diagonal $AC$ at an interior point $B_1$ such that $DB_1 \geq DA_1$. Prove that the quadrilateral $ABCD$ is cyclic if and only if
$$
\frac{BC}{DA_1} + \frac{BA}{DC_1} = \frac{AC}{DB_1}
$$
The segment $A_1C_1$ meets the diagonal $AC$ at an interior point $B_1$ such that $DB_1 \geq DA_1$. Prove that the quadrilateral $ABCD$ is cyclic if and only if
$$
\frac{BC}{DA_1} + \frac{BA}{DC_1} = \frac{AC}{DB_1}
$$