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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 isfor instancehas 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



