Circles at Coordinates Construction
Name: Ted M.
Two intersecting lines (L1, L2) are given. A point on each line is given (P1
on L1; P2 on L2). The points are NOT equidistant from the intersection of L1 and L2. How
can you construct two circles (C1, C2) meeting the following criteria:
1. C1 tangent to L1 at P1.2. C2 tangent to L2 at P2.3. The circles shall have a common
tangent at a third point, P3.
I do not know how to locate P3.
I think there must be many solutions, but I cannot figure out how to really get started.
I have tried simplifying by making L1 and L2 the x and y axes, and I have tried thinking
about it geometrically (creating triangles and sections of circles) and algebraically,
but I cannot figure out how many unknowns I really have (ie how many equations I need to
I think I have found a resource from which you can find the answers to your inquiry;
however, because it involves geometric constructions, it would not be feasible or
understandable to translate into text here. The reference is: CRC Concise Encyclopedia of
Mathematics, edited by Eric W. Weisstein. Look up "Circle Tangents"
Let us see...how do I give you enough of a start to get you going, without giving you
enough to take all the fun out of the problem?
Draw a line perpendicular to L1 and intersecting L1 at the point P1. The center of circle
C1 (let's just call this point C1) must be somewhere on this line if the circle is to be
tangent to L1 at P1. The same sort of thing is true for C2.
The point P3 must be somewhere on the line that goes from the point C1 to the point C2, in
order for both circles to have a common tangent at P3.
The distance from C1 to P1 must be the same as the distance from C1 to P3. Let us call this
distance R1, because it is just the radius of the circle. Similarly, the distance from C2
to P2 must be the same as the distance from C2 to P3, and I will call this distance R2.
There are an infinite number of solutions: for each value of R1, there is a single value of
R2 that satisfies the conditions.
You must think geometrically. To define a circle, you need a center and a radius. You
already have the centers. To get the radii (R1,R2), you need the perpendicular distance
from a point to a line. Find this for each point.
To have a common tangent point, the circles must touch at only one point. This is not
always true. It may be true for the given points. Also, it can happen if only one point
(P1) is given and you have to find P2 by yourself. In either case, the distance between P1
and P2 must be the sum of the two radii. If so, the common tangent occurs along the line
between P1 and P2 such that P3 is R1 from P1 and R2 from P2. If the points are right, the
problem can be done.
Dr. Ken Mellendorf
Illinois Central College
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Update: June 2012