Circles tangent to a given line and
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.
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
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
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.
Return to D. Hembree's
EMAT 6680 page
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