Dear Antreas,

I have got a very nice first result:

- Q is fixed and is incenter I

- Line is OH

The locus (or at least all points on this line) is line connected mittenpunkt and Gergonne point.

So mittenpunkt is NO VERY SPECIAL CASE.

Best regards,

Bui Quang Tuan

Quang Tuan Bui <

bqtuan1962@...> wrote:

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 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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