[Math] Do two new special points in any triangle exist

Question

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?

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:

• Triangle centers

Answer 1

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