In the hyperbolic plane, suppose $L$ is a line with points $A, B, C$ on the same side of line $L$. Assume further points $P, Q, R$ are on line $L$ such that segments $AP$, $BQ$ and $CR$ are perpendicular to $L$ and the three lengths $AP$, $BQ$ and $CR$ are all equal. Prove $A, B, C$ are not collinear.
Suppose they were collinear. Then $ACRQ$ is a rectangle since all the angles are $90$ degrees due to being perpendicular to the line Rectangles do not exist outside of Euclidean geometry since the interior angle sum is less then $360$ degrees.
Would that be sufficient to show its a rectangle? Or is there a better approach to take. Or is there any direct proof ideas that could work