Given a circular arc defined by three points $Ax$, $Ay$, $Bx$, $By$, $Cx$, $Cy$ (start, mid, end), is it possible to split it into two halves, i.e. to find the coordinates of the point $Mx$, $My$ so that $\|AM\|=\|MC\|$, using only basic arithmetic operations ($+, -, \times, \div$)? Or is it necessary to use some higher operations, such as square root or sine/cosine?
2026-04-01 17:55:28.1775066128
Circular arc bisection
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This point can be constructed with ruler and compass (find the center of the circle by intersecting two mediatrices, then find the middle of the chord and draw the radius through it).
This means that it can be solved with square roots.
Now, if you consider the arc through $(-2,0)$, $(-1,1)$ and $(2,0)$, its middle point is at $(0,\sqrt5-1)$. This should convince you that a square root is required.