Let L1,L2 be two lines intersected at A, and P a point. To draw line L intersecting L1,L2 at Ab,Ac, resp. such that:
- the circumcenter of AAbAc
Ab, Ac are the other than A intersections of the circle (P,PA) with the lines L1,L2
- the orthocenter of AAbAc
The perpendiculars to L1,L2 through P intersect L2,L1 at Ac,Ab, resp.
In general, P be a fixed point on the Euler line of AAbAc.
(ie PO/PH = m/n, where m,n given numbers)
Let ABC be a triangle and AaBbCc the orthic triangle. Let M1 be a line intersecting AB,AC at Ab,Ac resp. such that Aa is a fixed X(i) point on the Euler line of AAbAc. Similarly M2 a line intersecting BC,BA at Bc,Ba resp. such that Bb is X(i) point on the Euler line of BbBcBa, and M3 a line intersecting CA,CB at Ca,Cb such that Cc is X(i) point on the Euler line of CcCaCb.
For which points X(i) the Euler Lines of the triangles:
1. AAbAc, BBcBa, CCaCb
2. AaAbAc, BbBcBa, CcCaCb
[For X(i) = O is the Problem Concurrent Euler Lines]
APH, 22 February 2012