Given the triangle in black shown below:
Where the point at B is at the center of the circle, and the point at A is on the circle, and all points and angles A,B,C,a,b,c are known how can the points and angles be calculated for the red triangle which is changed by subtracting an amount from the angle C.
I see how it could be done with a ray intersection with the circle to find point A1, but it seems to me that there should be a simpler way.
Law of sines.
$\frac {\sin (C_1)}{ BA_1} = \frac {\sin(C_1)}{c}=M$ is a known value.
So $\frac {\sin \angle A_1}{a} = M$. Thus $m\angle A_1$ can be known.
And $ \angle A_1BC = 180 - m\angle C_1 - m\angle A_1$.
And the length $A_1C$ con be calculated as $\frac {\sin \angle A_1BC}{A_1C} = M$.
And the coordinates of $A_1$ can be calculated via $\sin$ and $\cos$ of $\angle A_1 BC$.