Dear Antreas,

Very nice idea! I try only some special case and we should do together to get some general results.

I am trying with first case: Q is fixed, and line is OH. I see the locus may be nice: one line?

The mittenpunkt may be is one very special case: Q = incenter I and the locus is only one point: mittenpunkt or some points?

These conjectures may be the good reasons for our efforts.

Best regards,

Bui Quang Tuan

PS: Please note that these only conjectures. I can confirm only with my first mittenpunkt message.

Antreas P. Hatzipolakis <

xpolakis@...> wrote:

On 1-07-06, Quang Tuan Bui wrote (partly):

> Given triangle ABC, mittenpunkt Mp, incenter I. One line passing

> through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.

> Similarly define Bc, Ba, Ca, Cb.

> 4. The most interesting:

> Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent

> at one point P.

Dear Tuan

We have here two interesting locus families

Let ABC be a triangle, and Q,P two points.

The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.

Similarly Bc,Ba, and Ca,Cb.

Which is the locus 1. of P 2. of Q

such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa

are concurrent.

Case 1 : Q = Fixed Point, P = Variable Point

Case 2 : Q = Variable Point, P = Fixed Point

Special Cases : P, or Q = I,O,H,G,K, etc

And of course we can ask for concurrence of other than OH lines

Antreas

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