# How to Construct Tangents from a Point to a Circle

Given circle $A(B)$ and point $C$ outside the circle, construct the tangents from $C$ to $A(B).$

Solution

Given circle $A(B)$ and point $C$ outside the circle, construct the tangents from $C$ to $A(B).$

The construction includes drawing an auxiliary circle on $AC$ as a diameter. The applet below illustrates the process.

Let $D$ be the midpoint of $AC$, then the circle in question is centered at $D$, passes through $A$ and $C$, and will be denoted $(D)$. Circle $(D)$ meets the given circle $A(B)$ in points $E$ and $F$. The angles $AEC$ and $AFC$ are inscribed in $(D)$ and subtend diameter $AC$; both are, therefore, right.

But $AE$ and $AF$ are radii of $A(B)$, and the line perpendicular to a radius of a circle at its end point is tangent to the circle. It follows that $CE$ and $CF$ are the sought tangents.

As a bonus, this construction shows how to construct a circle centered at a given point and orthogonal to a given circle. Indeed, cicle $C(E)$ fits the bill, because it passes through $E$ and $F$ and has $AE$ and $AF$ as tangents.

Note that when point $C$ is located on the radical axis of two circles, then a circle with center at $C$ orthogonal to one of the given circles is automatically orthogonal to the other.

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