Theorem 4.10: If a point on a circle is reflected about a line that goes through the center of the circle, then the image of the point is also on the circle.
Proof: If the point on the circle is on the line of reflection, then it is fixed by the reflection and so its image is also on the circle.
Let A be a point on the circle, and let O be the center of the circle. If A is not on the line of reflection, then by Theorem 3.6, the distance to its foot in the line of reflection is less than the distance from A to the center of the circle, so the foot of A in the line of reflection is inside the circle. Since the line from A to its foot in the line of reflection crosses the line of reflection inside the circle, by Theorem 3.8, the line from A to its foot is not tangent to the circle, and by Theorems 3.2 and 3.3, the line from A to its foot intersects the circle at two points. Call the other point B.
By Theorem 3.9, the midpoint of the line segment from A to B is the foot of the center of the circle in that line segment. By the uniqueness of a line through a point perpendicular to a given line, Theorem 1.8, the line of reflection is the line from the center of the circle to the midpoint of the line segment from A to B. Since the foot of the point in the line of reflection is halfway between A and B, we conclude that B, which is on the circle, is the reflection of A.