**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*.