If there is a point on a chord of a circle, which point on the minor arc would make the length between the the two points the smallest?

Question

If segment mn is a diameter of a circle f, and v is a fixed point on mn, for which point E on MN is the length FE minimized?

Answer 1

Yes. The chord is just a distraction. For any point $D$ inside a circle (other than the centre), the closest point to $D$ on the circle is the point where the radial line from the centre through $D$ meets the circle.

Answer 2

You are right. A proof would be: Let $F$ be any other point on the arc (different from our candidate $E$). Let $O$ be the center of the circle. Then we have $$ OD+DE=radius $$ whereas $$ OD+DF>OF=radius $$ (by the triangle inequality). Therefore, $DE<DF$.