# For what value on X provides the maximum angle α

In the figure below, what value on X provides the maximum angle α?

• How does the image function? Which values or lines/angles are movable and how does it affect anything else? Finally, what do you think or what is your plan? – Christopher Marley May 13 '18 at 19:48
• Is that supposed to be a straight line on the right-hand side? As drawn, there's a small "step" where the segment labeled $x$ intersects with the RHS. – Bungo May 13 '18 at 19:55
• My bad, there should not be an end to the right triangle. α's point is moving along the hypotenuse while maintaining the distance 1. – W. Emanuel May 13 '18 at 20:02

The vertex of angle $\alpha$ (point $G$ in diagram below) is constrained on a line parallel to $BF$ and vertically shifted by $1$. If $r$ is the radius of circle $GCD$, then $\sin\alpha=8/r$ and $\alpha$ gets then its maximum when the radius of the circle is minimum, that is when circle $GCD$ is tangent to the line of $G$.
From this geometric condition it is not difficult to find $r$ and then $\alpha$.