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Topic: a condition for regularity of a pentagram inscribed in a circle
Replies: 0

 Lee Rudolph Posts: 13 Registered: 12/6/04
a condition for regularity of a pentagram inscribed in a circle
Posted: Oct 29, 1993 3:13 PM

I thought up the following problem for a prize exam about 16 years ago.
(I don't claim it wasn't known before; that's just when it occured to
me.)

None of the contestants answered it. A few years later I put it on
another prize exam, and Rennie Mirollo gave an elegant solution (which,
alas, I have since forgotten).

THEOREM. Given five points on a circle (say, A, B, C, D, and E in
circular order), the pentagram with those vertices (that is, the figure
with edges AC, CE, EB, BD, and DA) is regular (that is--for
instance--has all five angles ACE, CEB, EBD, BDA, and DAC equal)
if and only if its five triangular "tips" all have the same area
(figure out what I mean by tips by drawing the picture).

Lee Rudolph