## Re: Cevian Trace Equal Area Points

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• Dear Francois thank you very much for explaining what is equicenter E and areal point Q. Another simple construction for Q comes from the following property:
Message 1 of 18 , Jul 31, 2007
Dear Francois
thank you very much for explaining
what is equicenter E and areal point Q.
Another simple construction for Q comes from the
following property:

If A'B'C', A"B"C" are two inscribed triangles
in ABC we construct the line La that passes
through A and is parallel to the line that connects
the mid points of B'C", B"C' and similarly
construct the lines Lb, Lc. These three lines
are concurrent at Q the areal point such that
the signed area (QA'A") = (QB'B") = (QC'C").

If in barycentrics the points are
A' = (0 : p : 1-p)
B' = (1-q : 0 : q)
C' = (r : 1-r : 0)
A" = (0 : p' : 1-p')
B" = (1-q' : 0 : q')
C" = (r' : 1-r' : 0)
then the point Q is the isotomic conjugate
of (p-p' : q-q' : r-r') and the signed area is
(ABC).(p-p').(q-q').(r-r')/K where
K = pq+qr+rp+p'q'+q'r'+r'p'+pp'+qq'+rr'-
-(p+q+r)(p'+q'+r').

Best regards

> Given any 2 inscribed triangles PaPbPc and UaUbUc of
> the reference triangle
> ABC, (cevian triangles are not needed!), there
> exists (in general), a unique
> point Q in the plane, such that:
>
> Area(QPaUa) = Area(QPbUb) = Area(QPcUc)
>
> In this formula , area means signed area of course!
>
> Now I give you a way (without proof for lack of
> room) to construct this
> point.
> 1° Let E be the fixed point of the affine map f
> sending PaPbPc on UaUbUc.
> E is the equicenter, following Neuberg notation. Why
> equicenter? Simply
> because E has same barycentrics in PaPbPc and
> UaUbUc.
> There is a very known construction of E, due to
> talked about it in some previous posts.
>
> 2° Let g be the affine map sending PaPbPc on ABC,
> then Q = g(E).
> It is also very easy to have a construction of Q.
>
> This area equality is the reason why I have called Q
> the areal center in
> choo-choo theory.

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• Anh Tuan oi Now, I criticize your formula: We construct point Q = X(6)*c(P)*c(U)*t(cd(P, U)) I find it too much complex for X(6) is not needed. You deduct it
Message 2 of 18 , Aug 1, 2007
Anh Tuan oi

We construct point Q = X(6)*c(P)*c(U)*t(cd(P, U))

I find it too much complex for X(6) is not needed.

You deduct it from the formula:

a). Barycentrics of Q = (q + r)*(v + w)/(q*w - r*v) : :

I look at ETC glossary the meaning of the point:
M = q*w - r*v::
where P = (p:q:r) and Q = (u:v:w)
M is the cross difference of P and Q only in case of using trilinears.
But here, you are using barycentrics so X(6) appareance is artificial though
needed to get a correct formula.

I don't find in ETC glossary the point M = q*w -r*v:: where P =(p:q:r) and Q
= (u:v:w) in using barycentrics;
M is simply the dual point of the line through P and Q.
Is there a special name for this point with such a projective definition?
Calling it AT(P,Q) for the moment, then we will have:
Q = c(P)*c(U)*t(AT(P,U))
Friendly
Francois

>
>
>

[Non-text portions of this message have been removed]
• Dear Francois and Nikos, Thank you very much for your interesting and general analyses from general view point. I learn a lot from your messages. In fact I
Message 3 of 18 , Aug 1, 2007
Dear Francois and Nikos,
Thank you very much for your interesting and general analyses from general view point. I learn a lot from your messages. In fact I construct point Q as following (some more complicated than Nikos):
PaPbPc, UaUbUc are two inscribed triangles in ABC
Rb = reflection of Ub in midpoint of APb
Rc = reflection of Uc in midpoint of APc
La = line connected A and midpoint of RbRc
Similarly construct Lb, Lc. Three lines La, Lb, Lc bound one triangle A'B'C'.
Two triangles ABC and A'B'C' are perspective at a point Q.
We can very easy to show that
Area(QPaUa) = Area(QPbUb) = Area(QPcUc)
Moreover:
Area(A'PaUa) = -Area(A'PbUb) = -Area(A'PcUc)
I have found this point and this construction by combine following small well known elementary locus problem into triangle:
Two segments XY, X'Y' are on given two lines. To find the locus of Z such that Area(ZXY) = Area(ZX'Y').
If areas are not signed then the locus is two lines passing intersection of two given lines. I apply this result for each vertex with two sidelines then I have got six lines as in my construction before.
From this construction we can easy to get barycentrics of Q. In order to get triangle center I choose two Cevian triangles of P, U and get compact expression:
Q = (q + r)*(v + w)/(q*w - r*v) : :
From this expression I see that Q = X(6)*c(P)*c(U)*t(cd(P, U)).
All another results I have got from this formula and by barycentric calculations.

Thank you again and best regards,
Bui Quang Tuan

---------------------------------
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• Dear friends, the point Q is on the line at infinity if the point (p-p : q-q : r-r ) is on the Steiner circumellipse. Hence the denominator K can be written
Message 4 of 18 , Aug 1, 2007
Dear friends,
the point Q is on the line at infinity if the
point (p-p' : q-q' : r-r') is on the
Steiner circumellipse.
Hence the denominator K can be written another way
K = xy+yz+zx where x=p-p', y=q-q', z=r-r'.
or
area = (ABC)/[1/(p-p')+1/(q-q')+1/(r-r')]
= (ABC)/(BC/A"A' + CA/B"B' + AB/C"C')

Best regards

> If A'B'C', A"B"C" are two inscribed triangles
> in ABC we construct the line La that passes
> through A and is parallel to the line that connects
> the mid points of B'C", B"C' and similarly
> construct the lines Lb, Lc. These three lines
> are concurrent at Q the areal point such that
> the signed area (QA'A") = (QB'B") = (QC'C").
>
> If in barycentrics the points are
> A' = (0 : p : 1-p)
> B' = (1-q : 0 : q)
> C' = (r : 1-r : 0)
> A" = (0 : p' : 1-p')
> B" = (1-q' : 0 : q')
> C" = (r' : 1-r' : 0)
> then the point Q is the isotomic conjugate
> of (p-p' : q-q' : r-r') and the signed area is
> (ABC).(p-p').(q-q').(r-r')/K where
> K = pq+qr+rp+p'q'+q'r'+r'p'+pp'+qq'+rr'-
> -(p+q+r)(p'+q'+r').
>
> Best regards

___________________________________________________________
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• Dear Tuan, ... Yes the same method I also used. If A is the intersection of XY, X Y and M, M are the mid points of XY , X Y then the locus such that the
Message 5 of 18 , Aug 1, 2007
Dear Tuan,

> I have found this point and this construction by
> combine following small well known elementary locus
> problem into triangle:
> Two segments XY, X'Y' are on given two lines. To
> find the locus of Z such that Area(ZXY) =
> Area(ZX'Y').
> If areas are not signed then the locus is two
> lines passing intersection of two given lines.

Yes the same method I also used.
If A is the intersection of XY, X'Y' and
M, M' are the mid points of XY', X'Y
then the locus such that the signed areas
are equal is the line parallel to MM' that
passes through A.

Best regards

___________________________________________________________
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• Dear Nikos and Tuan Thanks for your interesting remarks. Of course in choo-choo theory, these lines La, Lb, Lc play a central role and I have called them equal
Message 6 of 18 , Aug 1, 2007
Dear Nikos and Tuan
Of course in choo-choo theory, these lines La, Lb, Lc play a central role
and I have called them equal area axis.
Notice that their construction is affine! I had also my own construction
slightly different from yours.
As for Tuan formula, I would be happy to have a name if any for the dual
point M(q*w - r*v::)of the line through P(p:q:r) and Q(u:v:w) given by their
barycentrics.
Friendly
Francois
PS As Nikos notice, point isotomic of the areal center also plays an
important role in choo-choo theory.
As I go away from Paris in Britanny for several weeks even months far away
from the web, I give you a choo-choo configuration, so you can think about
me in this summer time:
Instead of cevian tiangles, I will look at pedal triangles PaPbPc and QaQbQc
of points P and Q wrt triangle ABC and I call L the line through P and Q.
Let E be the equicenter and S the areal center of the pedal triangles.
Then:
1° E is the orthopole of line L wrt ABC.
2° S is on the ABC-circumcircle and its Simson line is parallel to line L.

The first point was knew by Neuberg for a very very long time and maybe
that's why he found the orthopole. The first proof I saw was due to
V.Thebault
As for the second point, I would be happy to have some reference if any.
Of course these properties of points E and S are shared by any pair of
triangles of points on line L but this is another (choo-choo) story.

[Non-text portions of this message have been removed]
• Dear Nikos and Tuan ... So if abc and a b c are 2 inscribed triangles in ABC and if we use ordinary area ( i.e not signed), the Tuan-Nikos six lines cut
Message 7 of 18 , Aug 2, 2007
Dear Nikos and Tuan

> > Two segments XY, X'Y' are on given two lines. To
> > find the locus of Z such that Area(ZXY) =
> > Area(ZX'Y').
> > If areas are not signed then the locus is two
> > lines passing intersection of two given lines.
>
> I note this fact:
>
> These 2 lines of this locus are harmonic conjugate wrt lines XY and X'Y'.

So if abc and a'b'c' are 2 inscribed triangles in ABC and if we use ordinary
area ( i.e not signed), the Tuan-Nikos six lines cut themselves in 4
points forming an harmonic quad wrt ABC, each of these four points beeing an
equal area center wrt abc and a'b'c'. I will have to think about it in
choo-choo theory!
friendly
Francois

___________________________________________________________
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• Dear Nikos I give you my own construction of the equal area axis based on vectors. So you could compare it with yours. Given 4 points A, B, A , B such that
Message 8 of 18 , Aug 2, 2007
Dear Nikos
I give you my own construction of the equal area axis based on vectors.
So you could compare it with yours.
Given 4 points A, B, A', B' such that line AB and A'B' are through point O,
let L+ be the line locus of points M such that:
Area(MAB) = Area(MA'B')
and L- be the line locus of points M such that:
Area(MAB) = -Area(MA'B')
where Area means signed area.
Then L+ is the line through O directed by the vector V(AB) - V(A'B')
and L- is the line through O directed by the vector V(AB) + V(A'B').
That's why the four lines AB, A'B', L+, L- are in an harmonic range.
Friendly
Francois

On 8/2/07, Francois Rideau <francois.rideau@...> wrote:
>
> Dear Nikos and Tuan
>
>
>
> > > Two segments XY, X'Y' are on given two lines. To
> > > find the locus of Z such that Area(ZXY) =
> > > Area(ZX'Y').
> > > If areas are not signed then the locus is two
> > > lines passing intersection of two given lines.
> >
> > I note this fact:
> >
> > These 2 lines of this locus are harmonic conjugate wrt lines XY and
> > X'Y'.
>
>
> So if abc and a'b'c' are 2 inscribed triangles in ABC and if we use
> ordinary area ( i.e not signed), the Tuan-Nikos six lines cut themselves
> in 4 points forming an harmonic quad wrt ABC, each of these four points
> beeing an equal area center wrt abc and a'b'c'. I will have to think about
> it in choo-choo theory!
> friendly
> Francois
>
> ___________________________________________________________
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> >
> >
> >
> >
> >
>

[Non-text portions of this message have been removed]
• Dear Francois, [Tuan] ... [Francois] ... Yes. The line La+ is parallel to the segment of mid points of bc , b c and gives equal areas The line La- is parallel
Message 9 of 18 , Aug 2, 2007
Dear Francois,

[Tuan]
> > > Two segments XY, X'Y' are on given two lines.
> To
> > > find the locus of Z such that Area(ZXY) =
> > > Area(ZX'Y').
> > > If areas are not signed then the locus is two
> > > lines passing intersection of two given lines.
> >
> > I note this fact:
> >
> > These 2 lines of this locus are harmonic conjugate
> wrt lines XY and X'Y'.
>
[Francois]
> So if abc and a'b'c' are 2 inscribed triangles in
> ABC and if we use ordinary
> area ( i.e not signed), the Tuan-Nikos six lines
> cut themselves in 4
> points forming an harmonic quad wrt ABC, each of
> these four points beeing an
> equal area center wrt abc and a'b'c'. I will have to
> choo-choo theory!

Yes.
The line La+ is parallel to the segment of
mid points of bc', b'c and gives equal areas
The line La- is parallel to the segment of
mid points of bb', cc' and gives opposite areas
La+ and La- pass through A. Similarly consider
the lines Lb+, Lb-, Lc+, Lc-.
The lines La+, Lb+, LC+ are concurrent at Q
The lines La+, Lb-, LC- are concurrent at Qa
The lines La-, Lb+, LC- are concurrent at Qb
The lines La-, Lb-, LC+ are concurrent at Qc
If the areas we discuss (Paa'), (Pbb'), (Pcc')
are S1(P), S2(P), S3(P) then
S1(Q) = S2(Q) = S3(Q)
-S1(Qa) = S2(Qa) = S3(Qa)
S1(Qb) = -S2(Qb) = S3(Qb)
S1(Qc) = S2(Qc) = -S3(Qc)

Best regards

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• Dear Francois, I ve sent you a message and now I saw your message that you sent it before mine. As you see we use almost the same symbolism. You conclude the
Message 10 of 18 , Aug 2, 2007
Dear Francois,
I've sent you a message
and now I saw your message that you sent it
before mine.
As you see we use almost the same symbolism.
You conclude the harmonic relation using vectors
and I conclude it because my lines La+, La-, AB, AC
are parallel to the sides and diagonals of a
parallelogram.
Now I have an interesting result.
I investigated pedal triangles of
isogonal conjugate points.
If abc is the pedal triangle of P and
a'b'c' is the pedal triangle of P' isogonal
conjugate of P then the areal point Q of these
triangles lies on the circumcircle of ABC.
This is from a proof.
The following is from observation:
The point Q is the fourth intersection
of circumcircle with the rectangular circumhyperbola
of the circumcevian triangle of P, that passes
through P.
Similarly Q lies on the rectangular circumhyperbola
of the circumcevian triangle of P', that passes
through P'.

Best regards

> Dear Nikos
> I give you my own construction of the equal area
> axis based on vectors.
> So you could compare it with yours.
> Given 4 points A, B, A', B' such that line AB and
> A'B' are through point O,
> let L+ be the line locus of points M such that:
> Area(MAB) = Area(MA'B')
> and L- be the line locus of points M such that:
> Area(MAB) = -Area(MA'B')
> where Area means signed area.
> Then L+ is the line through O directed by the
> vector V(AB) - V(A'B')
> and L- is the line through O directed by the vector
> V(AB) + V(A'B').
> That's why the four lines AB, A'B', L+, L- are in an
> harmonic range.
> Friendly
> Francois

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• Dear Nikos Cheer up, I will look at your beautiful proof but notice that the areal point of any 2 pedal triangles is always on the circumcircle. I have several
Message 11 of 18 , Aug 2, 2007
Dear Nikos
Cheer up, I will look at your beautiful proof but notice that the areal
point of any 2 pedal triangles is always on the circumcircle.
I have several proofs of this fact and if you like calculus, one of them is
based on the Morley trick, i.e to choose the circumcircle as the unit circle
in the complex plane.
For other Hyacinthists, I give a sketch of my construction of the equal area
axis.
It is based in the following result of linear algebra:
S(L,M,N) = f(LM,LN)
where S is the signed area function , f is some determinant function that is
to say a bilinear skew symmetric function of 2 vectors and LM and LN beeing
vectors.
So if lines AB and A'B' are on O, we have:
S(M,A,B) = f(MA,MB) = f(MO + OA, MO + OB) = f(MO,OB) + f(OA,MO) = f(MO,OB -
OA) = f(MO,AB)
In the same way: S(M,A',B') = f(MO, A'B')
Now if k is any real, we have:
S(M,A,B) -k.S(M,A',B') = f(MO, AB -k.A'B') and we are done.
Friendly
Francois

>
> Dear Francois,
> I've sent you a message
> and now I saw your message that you sent it
> before mine.
> As you see we use almost the same symbolism.
> You conclude the harmonic relation using vectors
> and I conclude it because my lines La+, La-, AB, AC
> are parallel to the sides and diagonals of a
> parallelogram.
> Now I have an interesting result.
> I investigated pedal triangles of
> isogonal conjugate points.
> If abc is the pedal triangle of P and
> a'b'c' is the pedal triangle of P' isogonal
> conjugate of P then the areal point Q of these
> triangles lies on the circumcircle of ABC.
> This is from a proof.
> The following is from observation:
> The point Q is the fourth intersection
> of circumcircle with the rectangular circumhyperbola
> of the circumcevian triangle of P, that passes
> through P.
> Similarly Q lies on the rectangular circumhyperbola
> of the circumcevian triangle of P', that passes
> through P'.
>
> Best regards
>
> > Dear Nikos
> > I give you my own construction of the equal area
> > axis based on vectors.
> > So you could compare it with yours.
> > Given 4 points A, B, A', B' such that line AB and
> > A'B' are through point O,
> > let L+ be the line locus of points M such that:
> > Area(MAB) = Area(MA'B')
> > and L- be the line locus of points M such that:
> > Area(MAB) = -Area(MA'B')
> > where Area means signed area.
> > Then L+ is the line through O directed by the
> > vector V(AB) - V(A'B')
> > and L- is the line through O directed by the vector
> > V(AB) + V(A'B').
> > That's why the four lines AB, A'B', L+, L- are in an
> > harmonic range.
> > Friendly
> > Francois
>
>
>
>
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• Dear Francois and Nikos, If we choose two inscribed triangles as Cevian triangles of two Brocard points then Q can be one triangle center with barycentrics: Q
Message 12 of 18 , Aug 3, 2007
Dear Francois and Nikos,

If we choose two inscribed triangles as Cevian triangles of two Brocard points then Q can be one triangle center with barycentrics:
Q = 1/((a^4 - b^2*c^2)*(b^2 + c^2)) : :
Search value: -0.248703594444072
Q is on Steiner circumellipse, collinear with Brocard midpoint X(39) and Steiner point X(99).
Q is isotomic conjugate of X(732), one infinite point.

Have you all very nice weekend!
Best regards,
Bui Quang Tuan

Quang Tuan Bui <bqtuan1962@...> wrote: Dear All My Friends,
Given triangle ABC and two points P, U with barycentrics P = (p : q : r), U = (u : v : w). PaPbPc and UaUbUc are Cevian triangles of P, U respectively.
We construct point Q = X(6)*c(P)*c(U)*t(cd(P, U))
here:
c(P) = complement of P
c(U) = complement of U
t(cd(P, U)) = isotomic conjugate of cross difference of P, U
* = barycentric product
The results:
a). Barycentrics of Q = (q + r)*(v + w)/(q*w - r*v) : :
b). Area(QPaUa) = Area(QPbUb) = Area(QPcUc)
c). If P is on Lucas cubic then the locus of U such that Q is on circumcircle is also Lucas cubic.
d). If P = t(U) then the locus of U such that Q is on circumcircle is Lucas cubic (and three sidelines of antimedian triangle).
Best regards,
Bui Quang Tuan

---------------------------------
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[Non-text portions of this message have been removed]
• Dear Nikos It was not an answer to me, but a comment to me as list-owner. [ND] ... [FR] ... ... and was [ the post] incomplete APH -- [Non-text portions of
Message 13 of 18 , Aug 4, 2007
Dear Nikos

It was not an answer to me, but a comment to me as list-owner.

[ND]
>
>Dear Francois,
>You answered to Antreas, I think by mistake.
>I am not Antreas.

[FR]

> > Dear Antreas
> > Sorry, I send my post too soon!

... and was [ the post] incomplete

APH

--

[Non-text portions of this message have been removed]
• Dear Francois and Nikos, We denote S(T1, T2) = equal area point of two inscribed triangles T1, T2. Of course S(T1, T2) = S(T2, T1). Suppose T1, T2, T3 are
Message 14 of 18 , Aug 5, 2007
Dear Francois and Nikos,

We denote S(T1, T2) = equal area point of two inscribed triangles T1, T2. Of course S(T1, T2) = S(T2, T1).
Suppose T1, T2, T3 are three inscribed triangles then three equal area points S(T1, T2), S(T2, T3), S(T3, T1) are always on one circumconic. Denote this circumconic as C(T1, T2, T3).

Suppose now T1, T2, T3 are three Cevian triangles of P=X(2), U=X(4) and X respectively.
There is one special locus:
The locus of X such that circumconic C(T1, T2, T3) is passing also through X is one quintic:
CyclicSum[ (b^2 - c^2)*y*z*(a^2*SA*(x^3 - y*z*(y + z - x)) + b^2*c^2*(x^3 - y*z*x)) ] = 0
This circumquintic passes:
- Vetices of antimedial triangle.
- X(2), X(4), X(69), X(110), X(2574), X(2575)

Best regards,
Bui Quang Tuan

Quang Tuan Bui <bqtuan1962@...> wrote: Given triangle ABC and two points P, U with barycentrics P = (p : q : r), U = (u : v : w). PaPbPc and UaUbUc are Cevian triangles of P, U respectively.
We construct point Q = X(6)*c(P)*c(U)*t(cd(P, U))
here:
c(P) = complement of P
c(U) = complement of U
t(cd(P, U)) = isotomic conjugate of cross difference of P, U
* = barycentric product
The results:
a). Barycentrics of Q = (q + r)*(v + w)/(q*w - r*v) : :
b). Area(QPaUa) = Area(QPbUb) = Area(QPcUc)

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