Illustrate Monge's Circle Theorem: If the
possible 3 pairs of common exterior tangents to 3 circles exist and
intersect then the intersection points are collinear (lie on the same straight
Background & Techniques
I ran across this in Ivan Moscovich's excellent 2004 puzzle book
Leonardo's Mirror & Other Puzzles, which contains about 90 puzzles that
I haven't seen elsewhere. The link lists several used copies
available for under $5.00 and new copies for less than $8.00 both including
For the tangent lines to exist, no circle can be entirely contained within
another. For the 3 pairs of circle-to-circle tangent lines to
intersect, no 2 circles can have the same radius. The program
available here doesn't prove the theorem, but every apparent counter example i have found so far was the result of a program bug. Most of those
seem to have been squashed by now.
The program will define 3 random circles that meet the above conditions
(unique sizes and not completely overlapped) plus meeting the added
requirement that the intersections occur within the visible image area.
You may also use click and drag processing to define three circles.
The tangent lines are automatically drawn as circles are defined.
Once the circles are drawn by either method, they may be moved and
resized as desired to try to disprove the theorem (or break the program J).
Non-programmers are welcome to read on, but may
want to skip to the bottom of this page to download
executable version of the program.
preparatory update to our UGeometry unit added routines to solve the toughest math parts
of the problem (rotating and translating lines, circle-circle
intersections, point-circle tangents, and circle-circle external tangents).
Using those routines, mainly circle-circle
tangents, left the image drawing as the primary problem for this program to
solve. I decided to use a TPaintbox, control for drawing
efficiency. The Paint procedure redraws the figure each time it is
invoked. Setting the form doublebuffered property eliminates
flicker as the figures are being redrawn. The figures to be drawn are
- The circles (Circles:array[1..3] of TCircle;)
- The tangent lines, Tangents;array[1..6] of
- The line connecting the intersection points of
each of the 3 pairs of tangents, actually drawn as two lines,
MongeLine1, and MongeLine2, from each of the two most distant
intersection points to the 3rd point.
The "Create random circles" button creates a
random valid set of figures with tangents intersecting within the paint box
borders. The user may also use click and drag mouse operations to
define the three circles.
Circles created by either method may be moved or
resized using the mouse. Left mouse presses on or near the center of a
circle start a move operation. Presses near the circle edge and within the
circle start a resize operation. A "Mouse down" event exit determines
if the mouse is within an existing circle (to be moved or resized) or in an
empty area (defining a new circle). Every mouse move operation with
the left button pressed draws the all existing circles and tangents and the
line connecting the intersection points if they exist.
Two variables help the mouse exits determine
actions to be taken. Variable WorkingOn
identifies which circle is being moved or resized. Variable Working
specifies which operation (moving or sixing) is being carried out.
The program uses functions in our DFF Library
UGeometry unit. To recompile the program requires a one-time
download of our library file DFFLibV12.zip or later. .
Running/Exploring the Program
Suggestions for Further Explorations
Original Date: December 26, 2008
February 18, 2016