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Greedy Geoff sawed off a corner of a brick shaped block of Christmas cake, exposing a triangular fresh face of moist rich delicious gateau. He placed the tetrahedral fragment on the table, with its fresh face downwards. He mused through a port laden haze that it looked very stable, just like a mountain in fact, with its summit above a point inside its (not necessarily equilateral) triangular footprint $ABC$. He decided to decorate it, and took a UKMT pennant flying from a toothpick, and stuck it at the summit, with the flagpole perfectly vertical. Of course, the port was still at work and he is a bit clumsy, so he jammed the toothpick right through the cake, stabbing it into the tablecloth at a point $X$. Show that the circles $ABX$, $BCX$ and $CAX$ all have the same radius.

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    I thought gato meant cat.2012-11-22
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    http://en.wikipedia.org/wiki/Schl%C3%A4fli_orthoscheme2012-11-22
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    Damn it, contest problem: http://www.mathcomp.leeds.ac.uk/2012-11-22
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    Alright, evidently a public contest problem from 2005: see http://meta.math.stackexchange.com/questions/6629/another-contest-problem2012-11-23
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    Could you please make the title a little more descriptive?2012-11-23
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    But of course the OP should say (if he knows) "contest problem from 2005". Then Will (and others, like me) won't get all excited about it's appropriateness for this forum.2012-11-23

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