Let (C) be a circle, X,X' two antipodal points on (C), and P a point.
The centroid of the triangle PXX' is a fixed point.
The loci of the H's, O's, N's (and other points on the Euler Line) of
the triangles PXX', as X moves on (C), are lines perpendicular to PC.
Problem: Which is the locus of the Incenter I and the locus of the
Lemoine point K, as X moves on (C)?
Application in triangle ABC:
We can take a circle of ABC (circumcircle, incircle, NPC, etc)
and a point on the plane of ABC, and ask for loci of I's and K's
(and other centers).
Interesting is the case of P = O and (C) = NPC of ABC.
Let X,X' be two antipodal points on the NPC of ABC.
The triangles OXX' and ABC share the same centroid G.
Which is the locus of the incenter of OXX' as X moves on NPC?
Also which is the locus of the Lemoine point K?
Let ABC be a triangle.
A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.
[Equivalently: Let Ab, Ac be the orthogonal projections of B, C on L, resp.]
Let A* be the intersection of BAc and CAb.
Which is the locus of A* as L moves around A?
Parametric trilinear equation:
1/u(t) = a*((b^2+c^2-a^2)^2-4*b^2*c^2*c os(2*t)^2)/(2*S)
1/v(t) = 2*(cos(2*t)*c-b)*S - c*sin(2*t)*(a^2+3*b^2-2*cos(2* t)*b*c-c^2)
1/w(t) = 2*(cos(2*t)*b-c)*S + b*sin(2*t)*(a^2+3*c^2-2*cos(2* t)*b*c-b^2)