13486Re: [EMHL] Mittenpunkt And Concurrency Of Three Euler Lines
- Jul 1, 2006Dear 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.
Bui Quang Tuan
Quang Tuan Bui <bqtuan1962@...> wrote:
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.
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 passingDear Tuan
> 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.
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
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
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