Hello, dnoster!

I have a semicircle, and at certain intervals on the diameter, I have points.

I need to know the height from the points to the arc of the semicircle.

Code:

C
* * *
*/: \ *
* / : \ *
* / : \ *
/ :h \
*/ : \ *
*-----*--+--------*
A x P 2r-x B

A triangle inscribed in a semicircle is a *right* triangle: .$\displaystyle \angle C = 90^o$

We have a point $\displaystyle P$ on the diameter, where $\displaystyle AP = x.$

If $\displaystyle r$ is the radius of the circle, then $\displaystyle PB = 2r - x$

. . . . . . . . . . . . . . . . . **Theorem**

. . . . . The altitude to the hypotenuse of a right triangle

. . . . . . . . . . . . .is the mean proportional

. . . . . . . . of the two segments of the hypotenuse.

That is: .$\displaystyle h^2 \:=\:x(2r-x)$

Therefore: .$\displaystyle h\:=\:\sqrt{x(2r-x)}$