TITLE: how to find mid point of an arc?
QUESTION [7 upvotes]: I have start point $(x_1,y_1)$ and an end point $(x_2,y_2)$ and radius of arc. How to calculate the co-ordinates of mid-poing of arc? The arc is the part of a circle.

Known Values
length of AD // that is radius
B(x,y)
C(x,y)

Needs to find
D(x,y)  // D is the mid-point of arc BC

REPLY [3 votes]: First assume the reduced coordinates $C=(0,y),B=(0,-y),D=(x,0)$ so that $A=(x-r,0)$. Then we solve
$$\|AC\|^2=r^2=(x-r)^2+y^2$$ which gives the two solutions
$$x=r\pm\sqrt{r^2-y^2},$$ of which you will prefer the $+$ sign.

Now to deal with the general position, notice that $y=\dfrac{\|BC\|}2$ gives you $x$, and you add to the midpoint of $BC$ $x$ times the unit vector perpendicular to $BC$, i.e.
$$D=\frac{B+C}2+\left(r+\sqrt{r^2-\|BC\|^2}\right)\dfrac{BC}{\|BC\|}\times 1_z.$$