## Re: [EMHL] Mittenpunkt And Concurrency Of Three Euler Lines

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• ... 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
Message 1 of 9 , Jun 30 11:02 PM
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On 1-07-06, Quang Tuan Bui <bqtuan1962@...> 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
• 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
Message 2 of 9 , Jul 1, 2006
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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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• 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
Message 3 of 9 , Jul 1, 2006
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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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[Non-text portions of this message have been removed]
• Dear Antreas and All My Friends, Here are some results for the case: Q=incenter I and fixed, line is OH, P is any point on the line connected Gergonne point
Message 4 of 9 , Jul 1, 2006
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Dear Antreas and All My Friends,
Here are some results for the case:
Q=incenter I and fixed, line is OH, P is any point on the line connected Gergonne point and Mittenpunkt point. The concurrent point of three Euler lines as Pe.
(These results may be helpful for general case)
In the triangle PBcCb denote:
circumcircle as (Oa)
orthocenter as Ha,
altitude foot from Bc as B'c
altitude foot from Cb as C'b.
Similarly with another triangles:
PCaAc we have (Ob), Hb, C'a, A'c,
PAbBa we have (Oc), Hc, A'b, B'a.
Denote some intersections of lines:
Ah = AbA'b /\ AcA'c
Bh = BcB'c /\ BaB'a
Ch = CaC'a /\ CbC'b
Results:
1. Six points: B'c, C'b, C'a, A'c, A'b, B'a are on one conic (C1)
2. The circle (Oa) is tangent with line AbAc. Similarly for (Ob), (Oc). Other than P, these three circles cut each other at three point: Ap = (Ob) /\ (Oc), similarly define Bp, Cp. Denote circumcircle of ApBpCp as (Op).
3. Six points: Ha, Hb, Hc, Ah, Bh, Ch are on one conic (C2).
4. Pe is one intersection of circle (Op) and conic (C2).
Some special nice cases:
- When P is mittenpunkt then conic (C1) is a circle.
- When P is centroid then conic (C2) is a circle (Op).
Best regards,
Bui Quang Tuan

Quang Tuan Bui <bqtuan1962@...> wrote: 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 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

---------------------------------
Sneak preview the all-new Yahoo.com. It's not radically different. Just radically better.

[Non-text portions of this message have been removed]
• Dear Tuan [BQT] ... That is, Let ABC be a triangle, and P a point. The perpendicular to IA through P intersects AB,AC at Ab,Ac, resp. Similarly Bc,Ba, and
Message 5 of 9 , Jul 1, 2006
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Dear Tuan

[BQT]
> 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.

That is,

Let ABC be a triangle, and P a point.
The perpendicular to IA through P intersects AB,AC at Ab,Ac, resp.
Similarly Bc,Ba, and Ca,Cb.

The locus of P such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
are concurrent is Mittenpunkt-Gergone Line + ???

Probably the complete locus is some Cubic = Conic + Line

Nice result, indeed!

Antreas
--
• On 1-07-06, Quang Tuan Bui wrote: Dear Tuan ... Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line? APH
Message 6 of 9 , Jul 1, 2006
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On 1-07-06, Quang Tuan Bui <bqtuan1962@...> wrote:

Dear Tuan

> Here are some results for the case:
> Q=incenter I and fixed, line is OH, P is any point on the line
>connected Gergonne point and Mittenpunkt point. The concurrent point
>of three Euler lines as Pe.

> 4. Pe is one intersection of circle (Op) and conic (C2).

Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

APH
• Dear Antreas, The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it,
Message 7 of 9 , Jul 1, 2006
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Dear Antreas,
The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it, I can confirm.
Best regards,
Bui Quang Tuan

Antreas P. Hatzipolakis <xpolakis@...> wrote:
On 1-07-06, Quang Tuan Bui wrote:

Dear Tuan

> Here are some results for the case:
> Q=incenter I and fixed, line is OH, P is any point on the line
>connected Gergonne point and Mittenpunkt point. The concurrent point
>of three Euler lines as Pe.

> 4. Pe is one intersection of circle (Op) and conic (C2).

Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

APH

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• Dear Antreas, I can confirm: it is one line! Best regards, Bui Quang Tuan Quang Tuan Bui wrote: Dear Antreas, The locus of Pe as P moves
Message 8 of 9 , Jul 1, 2006
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Dear Antreas,
I can confirm: it is one line!
Best regards,
Bui Quang Tuan

Quang Tuan Bui <bqtuan1962@...> wrote:
Dear Antreas,
The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it, I can confirm.
Best regards,
Bui Quang Tuan

Antreas P. Hatzipolakis wrote:
On 1-07-06, Quang Tuan Bui wrote:

Dear Tuan

> Here are some results for the case:
> Q=incenter I and fixed, line is OH, P is any point on the line
>connected Gergonne point and Mittenpunkt point. The concurrent point
>of three Euler lines as Pe.

> 4. Pe is one intersection of circle (Op) and conic (C2).

Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

APH

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[Non-text portions of this message have been removed]
• Dear Tuan A variation of our locus is for Q = P That is Let ABC be a triangle, and P a point. The perpendicular to PA at P intersects AB,AC at Ab,Ac, resp.
Message 9 of 9 , Jul 1, 2006
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Dear Tuan

A variation of our locus is for Q = P

That is

Let ABC be a triangle, and P a point.
The perpendicular to PA at P intersects AB,AC at Ab,Ac, resp.
Similarly Bc,Ba, and Ca,Cb.

Which is the locus of P such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
are concurrent?

APH
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