Assignment #7

Construction of Two Circles Both Tangent to a Given Circle and a Line

By: Elizabeth Gore

To begin this investigation we will start out with a construction sequence yeilding two circles both tangent to a given circle and a line.

In this first sketch, I have simply constructed an arbitrary circle with K as the center and an arbitrary line, which I will make each of the two new circles tangent to.

The next step in construction is to form a line that will connect the circle and the line that the circles are to be tangent to.

If two new circles are to be tangent to the given circle at point P,

then the two new circles must be tangent to the line that is tangent to the given circle at point P.

To so this, I first constructed a line through the center of the circle K and the arbitrary point P.

I know this line will be perpendicular to the tangent line at P.

I have also marked the intersection of the tangent line and the given line at L.

Since each of the two new circles must be tangent to line PL at point L,

then it follows that their centers must lie on line KP (since the radius of each circle must be perpendicular to line PL at point P).

So where on line KP must these centers lie?

Since the new circles must be tangent to the given line, there also must be a radius that is perpendicular to that line.

Therefore, the center of each circle must be equidistant from the given line and from point P.

Here I have constructed a pair of angle bisectors of angles JLP and KLP, where J and K are simply points on either side of L that lie on the same line.

Where these bisectors cross line KP is where the centers(C1 and C2) of my new circles are.

BEHOLD, two circles both tangent to the given circle K and the given line containing point L.

Would you like a sketch to manipulate for yourself? If so click HERE.

If you would care to see the proof that C1 and C2 are really the centers of my two new circles click HERE.