PAUL PRIZE

Paul Erdos was a great mathematician who used to pose problems in mathematics periodicals (American Mathematical Monthly and others) with money prizes. I will do the same for a problem, but instead of money, the prize will be a classical book of geometry. It is a good coincidence that I have named the Point of the problem by the name of another great Paul: Pablo Picasso, and also today (18 January) is the birthday of my son Paul.

The Problem:

Let ABC be a triangle, A'B'C' the antipodal triangle of ABC (=circumcevian triangle of O) and D a point. Let A"B"C" be the circumcevian triangle of D with respect A'B'C' (ie A" is the other than A' intersection of A'D with the circumcircle etc)

Denote:

N,N1,N2,N3 the NPC centers of the triangles ABC, A"BC, B"CA, C"AB.

The four centers N,N1,N2,N3 are concyclic.

The center of the circle is General Picasso Point [See HERE]

Terms of Proofs:

The proof must be Euclidean synthetic (ie with no use of algebra or coordinate geometry)

The Prize:

The first, who will post a proof to the list Hyacinthos, will win the book:

F.G.-M.: Exercices de Geometrie. Huitieme Edition.

The book has 1302 pages + a 36 pages supplement

From page 1130 to page 1259: Geometrie du triangle:

Good Luck!

Point on the Euler Line

Let ABC be a triangle, A1B1C1 the pedal triangle of O = cevian triangle of G, G1G2G3 the circumcevian triangle of G with respect A1B1C1 (ie G1 is the other than A1 intersection of the NPC and AA1 etc), O1O2O3 the circumcevian triangle of O with respect A1A2A3 (ie O1 is the other than A1 intersection of A1O and the NPC etc) and GaGbGc the circumcevian triangle of O with respect G1G2G3 (ie Ga is the other than G1 intersection of OG1 and the NPC etc).

The triangles O1O2O3 and GaGbGc are perspective.

Perspector (on the Euler line of ABC) ?

APH 17 January 2012

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The perspector Q lies on the OH line and satisfies OQ:QH = -2 cosA cosB cosC

The coordinates are:

{(a^2 - b^2 - c^2) (a^4 - b^4 - 2 a^2 b c + 2 b^2 c^2 - c^4) (a^4 -
b^4 + 2 a^2 b c + 2 b^2 c^2 - c^4), -(a^2 - b^2 + c^2) (a^4 -
b^4 - 2 a b^2 c - 2 a^2 c^2 + c^4) (a^4 - b^4 + 2 a b^2 c -
2 a^2 c^2 + c^4), -(a^2 + b^2 - c^2) (a^4 - 2 a^2 b^2 + b^4 -
2 a b c^2 - c^4) (a^4 - 2 a^2 b^2 + b^4 + 2 a b c^2 - c^4)}

This is not in ETC.

ADDENDUM (9/9/19)

Peter Moses:
X(18531)

Generalization for point P instead of O:

P a point (instead of O)
A1B1C1 = the cevian triangle of G (medial triangle)
P1P2P3 = the circumcevian triangle of P with respect A1B1C1
G1G2G3 = the circumcevian triangle of G with respect A1B1C1
GaGbGc = the circumcevian triangle of P with respect G1G2G3

P1P2P3 and GaGbGc are perspective.
The perspector S is on line GP and satisfies the ratio:

k = GS:SP = (PN^2 - R^2/4) (a^2x+b^2 y+ c^2z)/(x+y+z), where N is the nine point center (and R/2 is the radius of the nine point circle)

Francisco Javier García Capitán
18 January 2012

PICASSO POINT Generalized

Generalization of PICASSO POINT.

Let ABC be a triangle, A'B'C' the antipodal triangle of ABC (=circumcevian triangle of O) and D a point.

Let A"B"C" be the circumcevian triangle of D with respect A'B'C' (ie A" is the other than A' intersection of A'D with the circumcircle etc)

Denote:

N1,N2,N3 the NPC centers of the triangles A"BC, B"CA, C"AB.

The four NPC centers N, N1,N2,N3 are concyclic.

Which is the center, General Picasso Point, of the circle ?

APH, 17 January 2012

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This is the midpoint of DH.

Francisco Javier García Capitán
17 January 2012

X56, X235

1. Let A1B1C1 be a triangle, A2B2C2 its antipodal triangle (=circumcevian of O), A3B3C3 the circumcevian triangle of I with respect the triangle A2B2C2 and A'B'C' the circumcevian triangle of I.

The triangles A'B'C', A3B3C3 are perspective.
Perspector (on the OI line) ?

2. Orthic Triangle Version:

Perspector (on the Euler line of ABC) ?

APH, 16 January 2012

1. X56
2. X235

Francisco Javier García Capitán
16 January 2012

PICASSO POINT [X(4550]

Let ABC be a triangle, A1B1C1 the pedal triangle of H, A2B2C2 the antipodal triangle of A1B1C1 with respect the circumcircle of A1B1C1 (= NPC of ABC), A'B'C' the antipodal triangle of ABC (= circumcevian triangle of O).

Denote:

A" = The other than A' intersection of A'A2 and the circumcircle of ABC

B" = The other than B' intersection of B'B2 and the circumcircle of ABC

C" = The other than C' intersection of C'C2 and the circumcircle of ABC

N1 = The NPC Center of A"BC

N2 = The NPC center of B"CA

N3 = The NPC center of C"AB

The four NPC centers N, N1,N2,N3 are concyclic (??).

Center of the circle ?

APH
14 January 2012

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Yes. The center is the point:

{a^2 (a^4 + a^2 b^2 - 2 b^4 + a^2 c^2 + 4 b^2 c^2 - 2 c^4) (a^4 -
2 a^2 b^2 + b^4 - 2 a^2 c^2 + 4 b^2 c^2 + c^4),
b^2 (-2 a^4 + a^2 b^2 + b^4 + 4 a^2 c^2 + b^2 c^2 - 2 c^4) (a^4 -
2 a^2 b^2 + b^4 + 4 a^2 c^2 - 2 b^2 c^2 + c^4),
c^2 (a^4 + 4 a^2 b^2 + b^4 - 2 a^2 c^2 - 2 b^2 c^2 + c^4) (-2 a^4 +
4 a^2 b^2 - 2 b^4 + a^2 c^2 + b^2 c^2 + c^4)}


The radius of the circle has a long expression:

-(1/(4 (p^2 - r^2 - 4 r R)^2))(3 p^6 - 5 p^4 r^2 + 5 p^2 r^4 -
3 r^6 - 36 p^4 r R + 56 p^2 r^3 R - 36 r^5 R - 46 p^4 R^2 +
224 p^2 r^2 R^2 - 190 r^4 R^2 + 368 p^2 r R^3 - 560 r^3 R^3 +
216 p^2 R^4 - 952 r^2 R^4 - 864 r R^5 - 324 R^6)

where p= semiperimeter.

Francisco Javier García Capitán
15 January 2012

Now in ETC X(4550)
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I name this new point as PICASSO Point in honor of great Pablo Picasso, whose a painting was recently stolen from the Greek National Gallery. See HERE and also my article in my blog Madara HERE

Antreas

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This point P lies on line G-X74 and satisfies the ratio

GP:PX74= (2 p^2 - 2 r^2 - 8 r R - 9 R^2)/(9 R^2).

Francisco, Hyacinthos 20683

Like Segovia point, also lies in line H-X1209, moreover Picasso point is the midpoint of H and Segovia point.

Francisco, Hyacinthos 20684

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The four nine-point centers are concyclic on a circle with center

(a^2(S^2+3S_{AA})(S^2+3S_{BC}) : ... : ...)

in barycentric coordinates.

This point lies on the line joining the centroid to X(74), the fourth intersection of the circumcircle with the Jerabek hyperbola.

Paul Yiu

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Denote:

Na = The NPC Center of AB"C"

Nb = The NPC center of BC"A"

Nc = The NPC center of CA"B"

The radical axes of the NPCs (N1) and (Na) - (N2) and (Nb) - (N3) and (Nc) are concurrent [See NPCs and Radical Axes], or equivalently the triangles N1N2N3, NaNbNc are perspective.

Antreas

SEGOVIA POINT

Let ABC be a triangle, A1B1C1 the pedal triangle of H, A2B2C2 the antipodal triangle of A1B1C1 with respect the circumcircle of A1B1C1 (= NPC of ABC) and A'B'C' the antipodal triangle of ABC (= circumcevian triangle of O)

The triangles A2B2C2, A'B'C' are perspective.

Perspector?

Generalization:

A1B1C1 = the pedal triangle of a point P (instead of H).

APH
14 January 2012

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For H the perspector is

{(a^2 - b^2 - c^2) (3 a^8 - 8 a^4 b^4 + 4 a^2 b^6 + b^8 +
4 a^4 b^2 c^2 - 4 a^2 b^4 c^2 - 4 b^6 c^2 - 8 a^4 c^4 -
4 a^2 b^2 c^4 + 6 b^4 c^4 + 4 a^2 c^6 - 4 b^2 c^6 +
c^8), -(a^2 - b^2 + c^2) (a^8 + 4 a^6 b^2 - 8 a^4 b^4 + 3 b^8 -
4 a^6 c^2 - 4 a^4 b^2 c^2 + 4 a^2 b^4 c^2 + 6 a^4 c^4 -
4 a^2 b^2 c^4 - 8 b^4 c^4 - 4 a^2 c^6 + 4 b^2 c^6 +
c^8), -(a^2 + b^2 - c^2) (a^8 - 4 a^6 b^2 + 6 a^4 b^4 -
4 a^2 b^6 + b^8 + 4 a^6 c^2 - 4 a^4 b^2 c^2 - 4 a^2 b^4 c^2 +
4 b^6 c^2 - 8 a^4 c^4 + 4 a^2 b^2 c^4 - 8 b^4 c^4 + 3 c^8)}

not in ETC

Locus:

circumcircle + Linf + cubic (I think not cathalogued in B. Gibert's Cubics in Triangle Plane)

The cubic is

-2 a^6 b^2 c^4 x^3 y + 4 a^4 b^4 c^4 x^3 y -
2 a^2 b^6 c^4 x^3 y + 2 a^2 b^2 c^8 x^3 y - a^8 c^4 x^2 y^2 +
2 a^6 b^2 c^4 x^2 y^2 - 2 a^2 b^6 c^4 x^2 y^2 + b^8 c^4 x^2 y^2 +
3 a^6 c^6 x^2 y^2 - a^4 b^2 c^6 x^2 y^2 + a^2 b^4 c^6 x^2 y^2 -
3 b^6 c^6 x^2 y^2 - 3 a^4 c^8 x^2 y^2 + 3 b^4 c^8 x^2 y^2 +
a^2 c^10 x^2 y^2 - b^2 c^10 x^2 y^2 + 2 a^6 b^2 c^4 x y^3 -
4 a^4 b^4 c^4 x y^3 + 2 a^2 b^6 c^4 x y^3 - 2 a^2 b^2 c^8 x y^3 +
2 a^6 b^4 c^2 x^3 z - 2 a^2 b^8 c^2 x^3 z - 4 a^4 b^4 c^4 x^3 z +
2 a^2 b^4 c^6 x^3 z + 9 a^6 b^4 c^2 x^2 y z -
5 a^4 b^6 c^2 x^2 y z - 5 a^2 b^8 c^2 x^2 y z + b^10 c^2 x^2 y z -
9 a^6 b^2 c^4 x^2 y z + 7 a^2 b^6 c^4 x^2 y z -
2 b^8 c^4 x^2 y z + 5 a^4 b^2 c^6 x^2 y z -
7 a^2 b^4 c^6 x^2 y z + 5 a^2 b^2 c^8 x^2 y z + 2 b^4 c^8 x^2 y z -
b^2 c^10 x^2 y z - a^10 c^2 x y^2 z + 5 a^8 b^2 c^2 x y^2 z +
5 a^6 b^4 c^2 x y^2 z - 9 a^4 b^6 c^2 x y^2 z +
2 a^8 c^4 x y^2 z - 7 a^6 b^2 c^4 x y^2 z +
9 a^2 b^6 c^4 x y^2 z + 7 a^4 b^2 c^6 x y^2 z -
5 a^2 b^4 c^6 x y^2 z - 2 a^4 c^8 x y^2 z - 5 a^2 b^2 c^8 x y^2 z +
a^2 c^10 x y^2 z + 2 a^8 b^2 c^2 y^3 z - 2 a^4 b^6 c^2 y^3 z +
4 a^4 b^4 c^4 y^3 z - 2 a^4 b^2 c^6 y^3 z + a^8 b^4 x^2 z^2 -
3 a^6 b^6 x^2 z^2 + 3 a^4 b^8 x^2 z^2 - a^2 b^10 x^2 z^2 -
2 a^6 b^4 c^2 x^2 z^2 + a^4 b^6 c^2 x^2 z^2 + b^10 c^2 x^2 z^2 -
a^2 b^6 c^4 x^2 z^2 - 3 b^8 c^4 x^2 z^2 + 2 a^2 b^4 c^6 x^2 z^2 +
3 b^6 c^6 x^2 z^2 - b^4 c^8 x^2 z^2 + a^10 b^2 x y z^2 -
2 a^8 b^4 x y z^2 + 2 a^4 b^8 x y z^2 - a^2 b^10 x y z^2 -
5 a^8 b^2 c^2 x y z^2 + 7 a^6 b^4 c^2 x y z^2 -
7 a^4 b^6 c^2 x y z^2 + 5 a^2 b^8 c^2 x y z^2 -
5 a^6 b^2 c^4 x y z^2 + 5 a^2 b^6 c^4 x y z^2 +
9 a^4 b^2 c^6 x y z^2 - 9 a^2 b^4 c^6 x y z^2 + a^10 b^2 y^2 z^2 -
3 a^8 b^4 y^2 z^2 + 3 a^6 b^6 y^2 z^2 - a^4 b^8 y^2 z^2 -
a^10 c^2 y^2 z^2 - a^6 b^4 c^2 y^2 z^2 + 2 a^4 b^6 c^2 y^2 z^2 +
3 a^8 c^4 y^2 z^2 + a^6 b^2 c^4 y^2 z^2 - 3 a^6 c^6 y^2 z^2 -
2 a^4 b^2 c^6 y^2 z^2 + a^4 c^8 y^2 z^2 - 2 a^6 b^4 c^2 x z^3 +
2 a^2 b^8 c^2 x z^3 + 4 a^4 b^4 c^4 x z^3 - 2 a^2 b^4 c^6 x z^3 -
2 a^8 b^2 c^2 y z^3 + 2 a^4 b^6 c^2 y z^3 - 4 a^4 b^4 c^4 y z^3 +
2 a^4 b^2 c^6 y z^3;

Francisco Javier García Capitán
14 January 2012

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I name the perspector for P = H as SEGOVIA Point for three reasons:

1. Segovia is Spanish as Francisco

2. His name is Andrés as mine

3. He was a great guitarist, and I am wishing to my daughter, who started recently lessons in guitar, to be like him!

Antreas

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This point S lies on line H-X1209 and satisfies the ratio:

HX1209:X1209S = (p^2 - r^2 - 4*r*R)/(3*p^2 - 3*r^2 - 12*r*R - 10*R^2).

Francisco, Hyacinthos 20678

Perspective

Let ABC be a triangle and OaObOc,HaHbHc the pedal triangles of O,H, resp.

Denote

A' = HaN /\ AO

B' = HbN /\ BO

C' = HcN /\ CO

Are the triangles OaObOc, A'B'C' perspective?

Generalization:

Let ABC be a triangle, P,P* two isogonal conjugate points, PaPbPc, P1P2P3 the pedal triangles of P, P*, resp. and M the midpoint of PP* (= the circumcenter of the common pedal circle of P amd P*).

Denote:

A' = P1M /\ AP

B' = P2M /\ BP

C' = P3M /\ CP

Which is the locus of P such that PaPbPc, A'B'C' are perspective?

APH, 13 January 2012

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The locus is a septic through I and H, and a nonic through I and O.

For P=I, H, O, the respective perpectors are I, H and X216.

Francisco Javier García Capitán
13 January 2012