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There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.

Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by geogebra sofware. I proposed a conjecture:

In any triangle exist two points $P$ so that: $AA'=BB'=CC'$.

My question: Is the conjecture above correct?

enter image description here

My geogebra:

The Red locus: If $P$ lie on red locus then $AA'=CC'$.

The Blue locus: If $P$ lie on red locus then $AA'=BB'$.

The Pink locus: If $P$ lie on pink locus then $CC'=BB'$

See also:

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    $\begingroup$ For equilateral triangle the number of such points is not 2 for sure $\endgroup$
    – Fedor Petrov
    Commented Sep 30, 2018 at 17:31

2 Answers 2

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Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$

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These two points are the focuses of Steiner circumconic https://en.m.wikipedia.org/wiki/Steiner_ellipse.

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