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Yiu's equilateral triangle-tripped points

My friend, mister Paul Yiu has sent to me a very new nice equilateral triangle associated with two Fermat points and Kiepert hyperbola of a reference triangle via email since 4 years ago. But I have no proof. This problem is very nice, so I post at here, I hope that have a solution.

  1. Let $ABC$ be a triangle with two Fermat points $F_1$, $F_2$. Circles with center $F_1$ radius $F_1F_2$ meets the Kiepert hyperbola again at three points $M$, $N$, $P$.

Then triangle $MNP$ perpestive to ABC at tripped points. This mean:

  • $MA$, $NB$, $PC$ are concurrent
  • $MB$, $NC$, $PA$ are concurrent
  • $MC$, $NA$, $PB$ are concurrent

Three points above collinear.

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  1. Let $ABC$ be a triangle with two Fermat points $F_1$, $F_2$. Circles with center $F_2$ radius $F_2F_1$ meets the Kiepert hyperbola again at three points $M$, $N$, $P$.

Then triangle $MNP$ perpestive to $ABC$ at tripped points. This mean:

  • $MA$, $NB$, $PC$ are concurrent
  • $MB$, $NC$, $PA$ are concurrent
  • $MC$, $NA$, $PB$ are concurrent

Three points above collinear