### Simson lines of Poncelet point of (ABCP)

Antreas Hatzipolakis (Hyacinthos #26266)

Let ABC be a triangle and P a point.

Denote:

Ta, Tb, Tc = the medial triangles of PBC, PCA, PAB, resp.

S = the point the circumcircles of Ta, Tb, Tc [= NPCs of PBC, PCA, PAB, resp.] are concurrent at (= the Poncelet point of (ABCP))

Sa, Sb, Sc = the Simson lines of S wrt triangles Ta, Tb, Tc, resp.

Which is the locus of P such that Sa, Sb, Sc are concurrent?

The entire plane?

**The Simson lines of S (= QA-P2 of quadrangle ABCP) wrt triangles Ta, Tb, Tc, resp. are concurrent for all P at QA-P6 of quadrangle ABCP.
**

From EQF (Chris van Tienhoven)
QA-P6

QA-P6: Parabola Axes Crosspoint

The Parabola Axes Crosspoint is the intersection point of the axes of the 2 parabolas that can be constructed through A,B,C,P

It is also is the Midpoint of the Euler-Poncelet Point QA-P2 and the Isogonal Center QA-P4.
Because these parabolas only can be constructed when the Reference Quadrangle is not concave a better definition of this point is: "the Midpoint of the Euler-Poncelet Point and the Isogonal Center".
Because the property related to the parabolas is much more appealing this point is named after its primary function.

It also can be reasoned that in a concave quadrangle this point represents the intersection point of the axes of the imaginary parabolas.

QA-P6 is the common point of the 4 Simson Lines of QA-P2 wrt the medial triangles of the component triangles ( QFG#548-550).

Angel Montesdeoca. Julio, 2017