<p>Is the sum of distances from two fixed ends of a diameter to every point on a circle always equal?
I know that for an ellipse, the sum of distances from the two focii to any point on the ellipse will always be equal. I know that a circle is a special ellipse, so I was trying to apply the same logic to a circle, except that the ends of the diameter of a circle are not the focii. There’s only one focus in a circle, which is the center.
Can someone elucidate?</p>
<p>It sounds like you have it sorted out already. You could think of a circle as an ellipse with zero as the distance between the two foci. So the sum of the distances is just 2 radii. But to answer your original question, “Is the sum of distances from two fixed ends of a diameter to every point on a circle always equal?”, the answer is no. Compare for instance what you get when you put one of the points at the end of a diameter with what you get when you put the point midway between the two endpoints. Just draw the picture and you will see right away those can’t possible be equal.</p>
<p>Thank you.</p>