Circles tangent to a given line and circle

by

D. Hembree

Assignment 7 investigates the topic of a circle tangent to two other circles and is a rich environment for exploration. I encourage you to start at the beginning and do all the activities. Click here to go to Dr. Jim Wilson's EMAT 6680 assignments page and look at assignment 7.

This page leads you through an exploration of one small part of assignment 7, part 14: Given a line and a circle with center K, take an arbitrary point P on the circle and construct two circles tangent to the given circle at P and tangent to the line.

A line and a circle K are given. The task is to construct two circles, each tangent to the given circle at an arbitrary point P and also tangent to the given line.

We proceed by assuming the problem solved and try to find geometric relationships that will locate the centers of the two desired circles.

Assume we have found circle C tangent to circle K at P and tangent to the given line at a point T.
We desire a geometric relationship that will locate either point C or point T, for if we can find one we will be led to the location of the other.

Consider the line tangent to circle C (and consequently circle K) at point P, intersecting our given line at a point E.
We now know how to find point T and point C.
Method 1:  ET = EP since the tangent segments to a circle from a point exterior to the circle are congruent, so the location of T is known relative to point E. A perpendicular to our given line at T intersects line KP at C.
Method 2: Since ET = EP, triangle PET is isosceles and the angle bisector of angle PET is also the perpendicular bisector of chord PT and therefore must contain the center of the desired circle C.

We will use method 1 and proceed as follows:

Draw line KP and construct the line perpendicular to KP at P, intersecting our given line at E. Construct circle E with radius EP, intersecting our given line at T. Construct the perpendicular at T, intersecting line KP at C. Circle C with radius CP (= CT) is one of our desired circles tangent to circle K at P and tangent to the given line.

We were asked to construct TWO circles tangent to circle K. Where is the second one? We've already done all the work necessary for its construction. Notice above that circle E intersects the given line at a second point that meets all of our requirements. We can produce a second tangent circle as before:

Circle R is tangent to circle K at P and to the given line at Q.

The purpose of this assignment was perhaps more than the completion of this relatively simple construction. Recall that point P on circle K was completly arbitrary. What are the consequences of changing its location? Click anywhere on the figure below to open Geometer's sketchpad and explore this situation dynamically by dragging point P.

Click the figure to open GSP and explore.

What is the locus of the centers of the two constructed circles? They appear to be parabolas. We can conclude that they are by a simple analysis of the situation as constructed.

Construct line SU parallel to the given line so that TS = KP. It is now clear that point C is equidistant from point K and from line SU, which is the focus-directrix definition of a parabola. The locus of C as P moves around circle K is a parabola.

A similar situation hold for point R. Construct a line parallel to our given line so that WQ = KP.
Then RW = RQ - WQ = RP - KP = RK
So R is equidistant from a point  and a line, again fitting the focus-directrix definition of a parabola.

Why was this assignment part of an investigation of circles tangent to other circles? Perhaps because the given line could be considered a "circle of infinite radius". Consider our given line replaced by Circle A as shown below.

The locus of point H is now a hyperbola, since HA - HK = (HQ+QA) - (HP+PK) = QA - PK, which is constant.
So H lies on a hyperbola with foci K and A, as you can explore by clicking the figure to open GSP.

As circle A becomes very large, we approach a limiting case where circle A can be considered to be the line given at the beginning of this assignment as shown below.

The locus of point H is still a hyperbola, but the limiting case as circle A becomes infinitely large is the situation discussed at the beginning of this exercise.