In a triangle $ABC$, let I be the incentre. Let $D$, $E$, $F$ be the intersections of $(ABC)$. with the lines through $I$ perpendicular to $BC$, $CA$, $AB$, respectively.

Define $O= BC \cap DE$ and $L= AC \cap DE$ . Define $IF\cap AB= R$ . Let $N=(BOF) \cap (LAF)$ .Prove that $N$,$R$,$F$ are collinear.

My progress: Since $F\in (ABC) $ , I thought of using simson points . So I took points $J$,$R$,$K$ as the simson points in $BC$,$BA$,$AC$ wrt point $F$ respectively. ( as shown in the diagram )

Then since $NBFO$ and $AFLN$ is cyclic, we get that $180- \angle ONF=\angle OBF=\angle CBF=180- \angle FAC=180 -\angle FAL = \angle FNL $ .

Hence points $O$,$N$,$L$ are collinear .

Now, I am stuck . I tried using phantom points but couldn't proceed . I am thinking of using Radical axis but still confused.

Here are some more observations which might be trivial but still, we have $BJFR$, $RFKA$,$CJFK$ concyclic . We also have $\Delta JFK \sim \Delta BFA $

Please post hints if possible.

Thanks in advance.

Ps: This is my own observation, so there is a very high chance that I might be wrong.

Below are a few diagrams for the problem.

enter image description here enter image description here

  • $\begingroup$ Yes, I just hided a few lines in geogebra . Lemme give a better diagram $\endgroup$ – Sunaina Pati Jul 30 '20 at 13:35
  • $\begingroup$ It is my own problem . Actually, It is inspired by another unanswered problem in MSE . $\endgroup$ – Sunaina Pati Aug 3 '20 at 1:59

This result is false! An IMOTCer said that when we use the relation tool, the geogebra shows that $N \not\in RF$ . enter image description here

Thank you show much to everyone for trying this problem .

  • 2
    $\begingroup$ Lol! But still this is what math , you make conjectures. sometimes it turns out to be true and sometimes false . Keep it up for your effort !. $\endgroup$ – Raheel Aug 6 '20 at 1:53
  • $\begingroup$ Thanks @Raheel ! $\endgroup$ – Sunaina Pati Aug 6 '20 at 1:55
  • 1
    $\begingroup$ @SunainaPati Looks like I committed the very mistake I told you to avoid. I thought I worked around it (by taking another pair of lines), but I can't reproduce it now. $\endgroup$ – Calvin Lin Sep 17 '20 at 12:43
  • $\begingroup$ @CalvinLin It's okay :) $\endgroup$ – Sunaina Pati Sep 18 '20 at 1:59

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