Traingles with common base

From news@mathforum.org  Fri Jul 26 10:43:09 2002
From: Steve Gray <stevebg@adelphia.net>
Subject: Traingles with common base
Date: Fri, 26 Jul 2002 13:11:56 +0000 (UTC)
To: geometry-puzzles@moderators.isc.org
To: geometry-puzzles@support1.mathforum.org

Draw an equilateral triangle Z0,Z1,Z2 and put point Y somewhere
inside it. This gives three triangles having Y as their "apex" angle.
Now take a separate line segment AB. Draw a triangle similar to Z0Z1Y
with Z0 on A and Z1 on B. Call the new vertex V0 which corresponds
to Y, Repeat with triangle Z1Z2Y, with Z1 on A and Z2 on B. Call the
new vertex V1 (on the same side of AB as V0). Repeat once more with
Z2Z0Y and call the new point V2, also on the same side of AB.

Show that V0V1V2 is equilateral and that its vertex does not move as
Y moves.

Now generalize this for a regular N-gon. The new points Vx form a
regular N-gon whose centroid is independent of Y.

This problem is original so far as I know. I am interested in the
simplest synthetic solution: no algebra, please. If this gets a
few solutions I will post another original problem that I think is
much harder.

Note added by RR: In the statement above, the assertion "its vertex does not move as Y moves" should be "its centroid does not move as Y moves".

This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 26, 2002.