Midpoint of ON

Let ABC be a triangle and L1,L2,L3 the external bisectors of the angles BOC,COA,AOB, resp. (they are parallels to BC,CA,AB, resp.)

Ab, Ac := the orthogonal projections of A on L2,L3, resp.

Bc, Ba := the orthogonal projections of B on L3,L1, resp.

Ca, Cb := the orthogonal projections of C on L1,L2, resp.

The Euler lines of AAbAc, BBcaBa, CCaCb are concurrent at
Q = Midpoint of ON.

Generalization:
P instead of O. Locus of P:
1. for ext. or int. bisectors of BPC, CPA, APC
2. for parallels through P to sidelines of ABC ?

APH, 16 December 2011