Find coordinates of points on circumference given segment height

Question

I have a circumference with radius $r$ and center in $O(cx, cy)$.

Then I know $h1$, $h2$ and $h3$ that are the distance from the top point to the points $a, b$, $g, f$ and $c, e$. How can I find the coordinates $x$ and $y$ of $a, b, c, e, f, g$?

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I try the idea of Alexey Burdin obtaining:

gx = cx + r * cos(r / (r - h2))
gy = h2

fx = cx + r * cos(r / (r - h2))
fy = h2

cx = cx + r * cos(r / (r - h3))
cy = h3

ex = cx + r * cos(r / (r - h3))
ey = h3

but it seems not to work because points are not in the right position.

Answer 1

If $(x, y) = (x, c_y + r -h)$ are the coordinates of one of these points then \begin{equation} r^2=(x-c_x)^2 + (y-c_y)^2 = (x - c_x)^2+(r-h)^2 \end{equation} Hence \begin{equation} x = c_x\pm \sqrt{r^2 - (r-h)^2} = c_x\pm\sqrt{h(2r-h)} \end{equation}