three disjoint circles A, B and C
of unequal radii situated entirely in each other's exterior, the common internal tangents of circles A and C
meet in Y, the common internal tangents of circles A and B meet
in Z and the common external tangents of circles B and C meet in
X. Then the points X, Y
and Z are collinear.
Instructions: Dynamic Geometry (Requires
Java 1.3 or higher and Java enable browser)
You can alter the figure above
dynamically in order to test and prove (or disproved)
conjectures and gain mathematical insight that is less readily
available with static drawings by hand. To explore this theorem use the
replay buttons above to move step by step (1-2-3-4-5) through the
the Start (Step 1),
to next break (Step 2-3-4-5),
To the end.
Manipulate the dynamic circles
by dragging the points
at any step.
The cursor keys (left, right,
up, down) move the picture. To give the keyboard focus to
the applet, click into it.
The + and - key change the size
This page contains a
C.a.R. interactive geometry applet by R. Grothmann. Please be
patient while the applet loads on your computer. If you are
using a dial-up connection, it may take a couple minutes. If you
get a warning-security asking 'Do you want to trust the
signed applet distributed by "Rene Grothmann"?'. Please
click 'Always', and you will not be troubled again.
If you can't see the presentation above, check a more recent
version of your browser. Alternately, it may be that your
browser supports Java, but that it’s currently set to disable
Java applets. Dynamic Geometry applet requires a Java Plug-in
1.3 or higher (More details at:
Miquel's Pentagram Dynamic Geometry
d'Alembert Three Circles Theorem I - Dynamic Geometry
Monge & d'Alembert Three Circles Theorem II - Dynamic