In a problem we have a triangle $[A_1BC]$ and we know that it is obtuse in $A_1$. Also $C'$ is the orthogonal projection of $C$ onto $A_1B$.
The original image does not have the circunference and I've shown the result using properties of angles in a circunference.
In short, it's easy to show that if $A_1$ is not in the segment $\left[C' B\right]$ then the angle will always be acute which contradicts the hypothesis.
Can you find another way of showing this result?
Thank you very much!