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Point $O$ lie inside $ABC$ triangle. Points $A1,B1,C1$ are projections of $O$ on heights led from $A,B,C$ Prove that if $AA1=BB1=CC1$ then $AA1=2r$, where $r$ is radius of circle inscribed in $ABC$ triangle. I 'd be happy with a hint that will lead me to answer

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Some hints:

  • there can be only one point $O$ such that $AA_1=BB_1=CC_1$
  • show that Nagel point has the desired property - it is quite easy if you know van Aubel's theorem (the second one)
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