Normal to Circle passes through Center

Theorem

A normal $\NN$ to a circle $\CC$ passes through the center of $\CC$.

Proof

Let $\CC$ be positioned in a Cartesian plane with its center at the origin.

Let $\NN$ pass through the point $\tuple {x_1, y_1}$.

From Equation of Normal to Circle Centered at Origin, $\NN$ has the equation:

$y_1 x - x_1 y = 0$

or:

$y = \dfrac {y_1} {x_1} x$

From the Equation of Straight Line in Plane: Slope-Intercept Form, this is the equation of a straight line passing through the origin.

As the geometry of a circle is unchanged by a change of coordinate axes, the result follows for a general circle in whatever frame.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $3$. The normal