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## Re: Orthologic Triangles - Locus

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• Dear Antreas 2. Let ABC be a triangle, A B C the pedal triangle of point P, A*B*C* the circumcevian triangle of P wrt A B C and A ,B ,C the second
Message 1 of 18 , Mar 16, 2013
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Dear Antreas

2. Let ABC be a triangle, A'B'C' the pedal triangle of point P,
A*B*C* the circumcevian triangle of P wrt A'B'C' and A",B",C" the
second intersections of the circles (P,PA*), (P,PB*), (P,PC*) with
the pedal circle of P, resp.

The locus of P such that the triangles ABC, A"B"C" are orthologic is the septic Qnnn:

-a^2 c^4 x^4 y^3 + b^2 c^4 x^4 y^3 + c^6 x^4 y^3 -
a^2 c^4 x^3 y^4 + b^2 c^4 x^3 y^4 - c^6 x^3 y^4 +
2 b^2 c^4 x^4 y^2 z - 2 a^2 c^4 x^2 y^4 z -
2 b^4 c^2 x^4 y z^2 - 2 b^4 c^2 x^3 y^2 z^2 +
2 b^2 c^4 x^3 y^2 z^2 + 2 a^4 c^2 x^2 y^3 z^2 -
2 a^2 c^4 x^2 y^3 z^2 + 2 a^4 c^2 x y^4 z^2 +
a^2 b^4 x^4 z^3 - b^6 x^4 z^3 - b^4 c^2 x^4 z^3 - 2 a^4 b^2 x^2
y^2 z^3 + 2 a^2 b^4 x^2 y^2 z^3 + a^6 y^4 z^3 - a^4 b^2 y^4 z^3 +
a^4 c^2 y^4 z^3 + a^2 b^4 x^3 z^4 + b^6 x^3 z^4 -
b^4 c^2 x^3 z^4 + 2 a^2 b^4 x^2 y z^4 - 2 a^4 b^2 x y^2 z^4 -
a^6 y^3 z^4 - a^4 b^2 y^3 z^4 + a^4 c^2 y^3 z^4=0,

(+ line at infinity (A'B'C' undefined triangle) + circumcircle (A', B' and C' points aligned) + a sextic ).

Points of the septic Qnnn:

A, B, C (triple points)
X(1) double , X(4)
excenters (which are double)
feet of the altitudes.

The sextic (without real points ???)is:

-b^2 c^4 x^4 y^2 - a^2 c^4 x^3 y^3 - b^2 c^4 x^3 y^3 +
c^6 x^3 y^3 - a^2 c^4 x^2 y^4 + a^2 b^2 c^2 x^4 y z -
b^4 c^2 x^4 y z - b^2 c^4 x^4 y z + a^2 b^2 c^2 x^3 y^2 z -
b^4 c^2 x^3 y^2 z + b^2 c^4 x^3 y^2 z - a^4 c^2 x^2 y^3 z +
a^2 b^2 c^2 x^2 y^3 z + a^2 c^4 x^2 y^3 z - a^4 c^2 x y^4 z +
a^2 b^2 c^2 x y^4 z - a^2 c^4 x y^4 z - b^4 c^2 x^4 z^2 +
a^2 b^2 c^2 x^3 y z^2 + b^4 c^2 x^3 y z^2 - b^2 c^4 x^3 y z^2 +
6 a^2 b^2 c^2 x^2 y^2 z^2 + a^4 c^2 x y^3 z^2 +
a^2 b^2 c^2 x y^3 z^2 - a^2 c^4 x y^3 z^2 - a^4 c^2 y^4 z^2 -
a^2 b^4 x^3 z^3 + b^6 x^3 z^3 - b^4 c^2 x^3 z^3 -
a^4 b^2 x^2 y z^3 + a^2 b^4 x^2 y z^3 + a^2 b^2 c^2 x^2 y z^3 +
a^4 b^2 x y^2 z^3 - a^2 b^4 x y^2 z^3 + a^2 b^2 c^2 x y^2 z^3 +
a^6 y^3 z^3 - a^4 b^2 y^3 z^3 - a^4 c^2 y^3 z^3 - a^2 b^4 x^2 z^4 - a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + a^2 b^2 c^2 x y z^4 -
a^4 b^2 y^2 z^4=0.

Best regards
Angel Montesdeoca

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
>
> 1. Let ABC be a triangle, A'B'C' the pedal triangle of point P
> and A",B",C" the second intersections of the circles (P,PA'),
> (P,PB'), (P,PC') with the pedal circle of P, resp.
>
> Which is the locus of P such that the triangles ABC, A"B"C"
> are orthologic?
>
> O is on the locus.
>
> 2. Let ABC be a triangle, A'B'C' the pedal triangle of point P,
> A*B*C* the circumcevian triangle of P wrt A'B'C' and A",B",C" the
> second intersections of the circles (P,PA*), (P,PB*), (P,PC*) with
> the pedal circle of P, resp.
>
> Which is the locus of P such that the triangles ABC, A"B"C"
> are orthologic?
>
> H is on the locus.
>
> Figures:
> http://anthrakitis.blogspot.gr/2013/03/orthologic-triangles-locus.html
>
> APH
>
• Let ABC be a triangle, P a point and A B C the pedal triangle of P. Denote: Bc = the orthogonal projection of B on PC B3 = the reflection of Bc in PB Cb =
Message 2 of 18 , Apr 29, 2014
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Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Bc = the orthogonal projection of B' on PC'
B3 = the reflection of Bc in PB'

Cb = the orthogonal projection of C' on PB'
C2 = the reflection of Cb in PC'

Similarly C1, A3 and A2, B1

Which is the locus of P such that ABC and triangle bounded by
(B3C2, C1A3, A2B1) are 1. orthologic 2. parallelogic?

For P = H
They are orthologic (the orth. center (ABC, (B3C2, C1A3, A2B1)) is (X(74))
and parallelogic (the parallelogic center (ABC, (B3C2, C1A3, A2B1)) is X(110)

APH
• Starting point: Let ABC be a triangle. Denote: La, Lb, Lc = the reflections of the Euler line in BC,CA,AB, resp. Oa, Ob, Oc = the reflections of O in La, Lb,
Message 3 of 18 , Nov 4, 2014
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Starting point:

Let ABC be a triangle.

Denote:

La, Lb, Lc = the reflections of the Euler line in BC,CA,AB, resp.

Oa, Ob, Oc = the reflections of O in La, Lb, Lc resp.

ABC, OaObOc are orthologic.
Which is the other than O orthologic center?
(ie the orthologic center (OaObOc, ABC))

Generalization:

Let ABC be a triangle and P a point.

Denote:

La, Lb, Lc = the reflections of the Euler line in BC,CA,AB, resp.

Pa, Pb, Pc = the reflections of P in La, Lb, Lc resp.

Which is the locus of P such that ABC, PaPbPc are orthologic ?

APH

• Let ABC be a triangle and P a point. Denote: A B C = the pedal triangle of P O = the circumcenter of A B C ( = the center of the pedal circle of P) A B C =
Message 4 of 18 , Dec 16, 2014
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Let ABC be a triangle and P a point.

Denote:

A'B'C' = the pedal triangle of P

O' = the circumcenter of A'B'C' ( = the center of the pedal circle of P)

A"B"C" = the pedal triangle of O'.

Which is the locus of P such that A'B'C', A"B"C"
are orthologic?

On the locus:

1. P = H (O' = N)

Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C'

Orthologic center (A'B'C', A"B"C") = ?

2. P = Ο (O' = N)

Orthologic center (A"B"C", A'B'C') = the O of A'B'C' =  the NPC center of ABC

Orthologic center (A'B'C', A"B"C") = ? on the Euler line of the orthic triangle
Which point is this wrt tr. ABC and which wrt orthic tr. of ABC?

This is interesting because we can take as reference triangle the orthic
tiangle in order to get a point on the Euler line of ABC.

That is:

Let ABC be a triangle and A'B'C' the antipedal triangle of I.
(ABC is the pedal triangle of H of A'B'C'').
Now:
Let A1B1C1 be the pedal triangle of N of A'B'C' [ = O of ABC]
Let A2B2C2 be the pedall triangle of O of A'B'C'

A1B1C1, A2B2C2 are orthologic.

The orthologic center (A1B1C1, A2B2C2) is N of A'B'C' [= O of ABC]
The orthologic center (A2B2C2, A1B1C1) lies on the Euler line
of ABC. Point ?

APH

• ... Circumcircle, infinity, K003 . McCay cubic 1) Orthologic center (A B C , A B C ) = the NPC
Message 5 of 18 , Dec 16, 2014
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[Antreas]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the pedal triangle of P

O' = the circumcenter of A'B'C' ( = the center of the pedal circle of P)

A"B"C" = the pedal triangle of O'.

Which is the locus of P such that A'B'C', A"B"C"
are orthologic?

On the locus:

1. P = H (O' = N)

Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C'

Orthologic center (A'B'C', A"B"C") = ?

2. P = Ο (O' = N)

Orthologic center (A"B"C", A'B'C') = the O of A'B'C' =  the NPC center of ABC

Orthologic center (A'B'C', A"B"C") = ? on the Euler line of the orthic triangle
Which point is this wrt tr. ABC and which wrt orthic tr. of ABC?

This is interesting because we can take as reference triangle the orthic
tiangle in order to get a point on the Euler line of ABC.

That is:

Let ABC be a triangle and A'B'C' the antipedal triangle of I.
(ABC is the pedal triangle of H of A'B'C'').
Now:
Let A1B1C1 be the pedal triangle of N of A'B'C' [ = O of ABC]
Let A2B2C2 be the pedall triangle of O of A'B'C'

A1B1C1, A2B2C2 are orthologic.

The orthologic center (A1B1C1, A2B2C2) is N of A'B'C' [= O of ABC]
The orthologic center (A2B2C2, A1B1C1) lies on the Euler line
of ABC. Point ?

APH

[Peter Moses]:

>Which is the locus of P such that A'B'C', A"B"C" are orthologic?
Circumcircle, infinity, K003. McCay cubic

1)
Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C' = X(143) of ABC
Orthologic center (A'B'C', A"B"C") = on ABCs lines {{4,93},{6,24},{25,195},...} = X(79) of Orthic?
2)
Orthologic center (A'B'C', A"B"C") = X(1209) of ABC, X(3651) of Orthic. (probably!)

Best regards
Peter

• ... ************************** Thanks, Peter, What a surprise! A new point in such a simple construction !!!! I wouldn t bet a cent for it as a new point :-)
Message 6 of 18 , Dec 16, 2014
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[Antreas]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the pedal triangle of P

O' = the circumcenter of A'B'C' ( = the center of the pedal circle of P)

A"B"C" = the pedal triangle of O'.

Which is the locus of P such that A'B'C', A"B"C"
are orthologic?

On the locus:

1. P = H (O' = N)

Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C'

Orthologic center (A'B'C', A"B"C") = ?

2. P = Ο (O' = N)

Orthologic center (A"B"C", A'B'C') = the O of A'B'C' =  the NPC center of ABC

Orthologic center (A'B'C', A"B"C") = ? on the Euler line of the orthic triangle
Which point is this wrt tr. ABC and which wrt orthic tr. of ABC?

[Peter Moses]:

>Which is the locus of P such that A'B'C', A"B"C" are orthologic?
Circumcircle, infinity, K003. McCay cubic

1)
Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C' = X(143) of ABC
Orthologic center (A'B'C', A"B"C") = on ABCs lines {{4,93},{6,24},{25,195},...} = X(79) of Orthic?
2)
Orthologic center (A'B'C', A"B"C") = X(1209) of ABC, X(3651) of Orthic. (probably!)

Best regards
Peter

**************************

Thanks, Peter,

What a surprise! A new point in such a simple construction !!!!
I wouldn't bet a cent for it as a new point :-)

Description of the point:

Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles of H,N, resp.

A'B'C', A"B"C" are orthologic with orthologic center (A'B'C', A"B"C")
on {4,93},{6,24},{25,195},.. lines of ABC.

If you compute other "things" of the point (coordinates etc) please let me know
(to inform Clark)

Season's Greetings

APH
• ... *************************************** Another observation in the same configuration. But let s first rename the pedal triangles. Let ABC be a triangle
Message 7 of 18 , Dec 16, 2014
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[Antreas]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the pedal triangle of P

O' = the circumcenter of A'B'C' ( = the center of the pedal circle of P)

A"B"C" = the pedal triangle of O'.

Which is the locus of P such that A'B'C', A"B"C"
are orthologic?

On the locus:

1. P = H (O' = N)

Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C'

Orthologic center (A'B'C', A"B"C") = ?

2. P = Ο (O' = N)

Orthologic center (A"B"C", A'B'C') = the O of A'B'C' =  the NPC center of ABC

Orthologic center (A'B'C', A"B"C") = ? on the Euler line of the orthic triangle
Which point is this wrt tr. ABC and which wrt orthic tr. of ABC?

[Peter Moses]:

>Which is the locus of P such that A'B'C', A"B"C" are orthologic?
Circumcircle, infinity, K003. McCay cubic

1)
Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C' = X(143) of ABC
Orthologic center (A'B'C', A"B"C") = on ABCs lines {{4,93},{6,24},{25,195},...} = X(79) of Orthic?
2)
Orthologic center (A'B'C', A"B"C") = X(1209) of ABC, X(3651) of Orthic. (probably!)

Best regards
Peter

***************************************

Another observation in the same configuration.
But let's first rename the pedal triangles.

Let ABC be a triangle and AhBhCh, AoBoCo, AnBnCn the pedal triangles
of H,O,N resp

The midpoint of the line segment joining the orthologic centers (AhBhCh, AnBnCn) and (AoBoCo, AnBnCn) is the orthocenter of AnBnCn.

Which is this point (the orthocenter of the pedal triangle of N)?

And is it a general property of all (pairs of isogonal conj.) points on the locus (the McCay cubic)?

ie that, that midpoint is the orthocenter of the pedal triangle of the center of the common pedal circle of the isog. conj. points?
(Or at least is it lying on the Euler line of the triangle)?

Hmmm....... neither I understand at first glance what I wrote ("the orthocenter
of the pedal triangle of the center of the common pedal circle of the isog. conj.
points") !  :-). So let me rewrit it

Let ABC be a triangle and P,P* two isogonal conjugate points on the McCay
cubic and let Q be the center of the common pedal circle of P,P* (= midpoint
of PP*)

Denote:
T1, T2, T3 = the pedal triangles of P,  P*, Q, resp.

Let Mp be the midpoint of the line segment joining the orthologcic centers (T1, T3) and (T2, T3).

Is Mp the orthocenter of T3 for all P's (on the cubic)?
Or in general, is it lying on the Euler line of T3 (orthocenter or not) ?

Which is its locus as P moves on the McCay cubic?

APH

• From: Randy Hutson Dear Antreas and Peter, This point appears in César E. Lozada s Perspective-Orthologic-Parallelogic.pdf as the orthologic center of orthic
Message 8 of 18 , Dec 16, 2014
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From: Randy Hutson

Dear Antreas and Peter,

This point appears in César E. Lozada's Perspective-Orthologic-Parallelogic.pdf as the orthologic center of orthic and reflection triangles, also reflection of X(54) in X(973), with trilinears a*((S^2-SA^2)*(SW-R^2)+2*S^2*SA)/SA : : and search value 12.481825032289.

This point is also the orthic isogonal conjugate of X(1594), and X(79) of orthic IF ABC is acute.

Best regards,
Randy

• Dear Antreas and Peter, This point appears in César E. Lozada s Perspective-Orthologic-Parallelogic.pdf as the orthologic center of orthic and reflection
Message 9 of 18 , Dec 16, 2014
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Dear Antreas and Peter,

This point appears in César E. Lozada's Perspective-Orthologic-Parallelogic.pdf as the orthologic center of orthic and reflection triangles, also reflection of X(54) in X(973), with trilinears a*((S^2-SA^2)*(SW-R^2)+2*S^2*SA)/SA : : and search value 12.481825032289.

This point is also the orthic isogonal conjugate of X(1594), and X(79) of orthic IF ABC is acute.

Best regards,
Randy

********************

Thanks, Randy!!

Yes!!!! since the pedal trangle of N and the reflection triangles are homothetic

Cesar's table published in
and later also appeared the point in question in the paper
Jesus Torres, The triangle of reflections,Forum Geometricorum, 14 (2014) 265--294  [in Theorem 4.3].
(published in 7 Oct. 2014)

APH

• ... [Peter Moses]: A few extra properties of the point: Barys: a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^4 b^2-2 a^2
Message 10 of 18 , Dec 16, 2014
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[Antreas]:

Let ABC be a triangle and P a point.

Denote:

A'B'C' = the pedal triangle of P

O' = the circumcenter of A'B'C' ( = the center of the pedal circle of P)

A"B"C" = the pedal triangle of O'.

Which is the locus of P such that A'B'C', A"B"C"
are orthologic?

On the locus:

1. P = H (O' = N)

Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C'

Orthologic center (A'B'C', A"B"C") = ?

2. P = Ο (O' = N)

Orthologic center (A"B"C", A'B'C') = the O of A'B'C' =  the NPC center of ABC

Orthologic center (A'B'C', A"B"C") = ? on the Euler line of the orthic triangle
Which point is this wrt tr. ABC and which wrt orthic tr. of ABC?

[Peter Moses]:

>Which is the locus of P such that A'B'C', A"B"C" are orthologic?
Circumcircle, infinity, K003. McCay cubic

1)
Orthologic center (A"B"C", A'B'C') = the NPC center of A'B'C' = X(143) of ABC
Orthologic center (A'B'C', A"B"C") = on ABCs lines {{4,93},{6,24},{25,195},...} = X(79) of Orthic?
2)
Orthologic center (A'B'C', A"B"C") = X(1209) of ABC, X(3651) of Orthic. (probably!)

Best regards
Peter

**************************

Thanks, Peter,

What a surprise! A new point in such a simple construction !!!!
I wouldn't bet a cent for it as a new point :-)

Description of the point:

Let ABC be a triangle and A'B'C', A"B"C" the pedal triangles of H,N, resp.

A'B'C', A"B"C" are orthologic with orthologic center (A'B'C', A"B"C")
on {4,93},{6,24},{25,195},.. lines of ABC.

If you compute other "things" of the point (coordinates etc) please let me know
(to inform Clark)

Season's Greetings

APH

[Peter Moses]:

A few extra properties of the point:

Barys: a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::

On lines {{4,93},{6,24},{25,195},{49,143},{52,539},{70,6145},{113,5446},{155,3060},{403,3574},{648,1179},{1112,2914},{1209,1216},{1843,5965},{1986,3575}}.
Reflection of X(i) in X(j) for these {i,j}: {54,973},{1493,143},{2914,1112}.
3 X[1209] - 2 X[1216].
3 X[54] - 5 X[3567].
6 X[973] - 5 X[3567].
X(4)-ceva conjugate of X(1594).
X(4)-crosspoint of X(3518).
X(3)-crosssum of X(3519).
X(2216)-isoconjugate of X(3519).
Trilinear product X(1594) X(2964).
Barycentric product X(1594) X(1994).
X(79) Orthic triangle.
{{1,21},{7,79},...} of Tangential triangle.

Best regards,
Peter.

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