# Do two new special points in any triangle exist?

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:

• For equilateral triangle the number of such points is not 2 for sure – Fedor Petrov Sep 30 '18 at 17:31

## 2 Answers

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

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

• Since a Google search for "Steiner circumconic" didn't return too convincing results, why not add its definition? – Alex M. Sep 30 '18 at 18:51