I am trying to draw the following part of circle on a mobile screen but I can't do the math properly.
Please, ignore the text.
The left top corner is (0, 0) and the right bottom corner is (w, h). The width of the circular part is w/2 which is half the screen width.
To acieve that I want to draw I a big circle out of screen and only a small part of it inside.
So, the required output variables are the center coordinates of the circle (x, y) and the circle radius.
Obviously the x coordinate will be negative and the y coordinate will be positive.
Fact: Apple devices have no screen width/2 equals to height. Observation: when the circular part width and height difference increases the circle radius increase exponantially.
Result of the accepted solution by Ethan Bolker:
Two different device screens:
In this picture
$PQ$ is the mean proportional between $AP$ and $PB$. That means $$ h^2 = \frac{w}{2}(2x + \frac{w}{2}) = wx + \frac{w^2}{4} $$ so $$ x = \frac{h^2 - w^2/4}{w}. $$
That should be enough information for you to compute the coordinates of $C$.