Normal to Circle passes through Center
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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