TITLE: Do two new special points in any triangle exist?
QUESTION [10 upvotes]: 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

REPLY [10 votes]: Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$

REPLY [7 votes]: These two points are the focuses of Steiner circumconic https://en.m.wikipedia.org/wiki/Steiner_ellipse.