Κυριακή, 29 Ιανουαρίου 2012

Perspective


Let ABC be a triangle, HaHbHc the cevian triangle of H (orthic triangle), GaGbGc the cevian triangle of G (medial triangle), H1H2H3, G1G2G3 the circumcevian triangles of H and G, resp.


Denote:

A1 = the 2nd intersection of H1Ga and the circumcircle (other than H1)
A2 = the 2nd intersection of G1Ha and the circumcircle (other than G1)

B1 = the 2nd intersection of H2Gb and the circumcircle (other than H2)
B2 = the 2nd intersection of G2Hb and the circumcircle (other than G2)

C1 = the 2nd intersection of H3Gc and the circumcircle (other than H3)
C2 = the 2nd intersection of G3Hc and the circumcircle (other than G3)

The lines A1A2, B1B2, C1C2 bound a triangle A*B*C*
(A* = B1B2 /\ C1C2, B* = C1C2 /\ A1A2, C* = A1A2 /\ B1B2)

The triangles ABC, A*B*C* are perspective.

Perspector?

Generalization:
Locus of P instead of 1.H, 2.G

APH 29 January 2012

-----------------------------------------------------------------

The coordinates are:

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

Not in ETC

The result is true for any P, Q.
The Isogonal Conjugate of the perspector for P=(u:v:w) and Q(x:y:z) is:.

{u x (w^2 y^2 - v w y z + v^2 z^2), v y (w^2 x^2 - u w x z + u^2 z^2), wz (v^2 x^2 - u v x y + u^2 y^2) }

Francisco Javier García Capitán
30 January 2012

Πέμπτη, 26 Ιανουαρίου 2012

LOCUS


Let ABC be triangle, P a point, A1B1C1 the cevian triangle of P and A2B2C2 the circumcevian triangle of P.


Denote:

Ab = the other than A2 intersection of the circumcircle and A2B1
Ac = the other than A2 intersection of the circumcircle and A2C1

Bc = the other than B2 intersection of the circumcircle and B2C1
Ba = the other than B2 intersection of the circumcircle and B2A1

Ca = the other than C2 intersection of the circumcircle and C2A1
Cb = the other than C2 intersection of the circumcircle and C2B1

The lines AbAc, BcBa, CaCb bound a triangle A3B3C3.

Which is the locus of P such that:

1. ABC, A3B3C3

2. A1B1C1, A3B3C3

3. A2B2C2, A3B3C3

are perspective?


APH, 26 January 2012

----------------------------------------------------

For 1. and 3. the locus is the whole plane.
For 2, the locus is (symmedians) + circumcircle + (12th degree curve whose isogonal conjugate is 6th degree curve).

For 1. the perspector is

{a^2 x (c^2 y + b^2 z) (c^4 x^2 y^2 + b^2 c^2 x^2 y z +
2 a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + a^2 c^2 x y^2 z +
b^4 x^2 z^2 + 2 a^2 b^2 x y z^2 + a^4 y^2 z^2),
b^2 y (c^2 x + a^2 z) (c^4 x^2 y^2 + 2 b^2 c^2 x^2 y z +
a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + a^2 c^2 x y^2 z +
b^4 x^2 z^2 + 2 a^2 b^2 x y z^2 + a^4 y^2 z^2),
c^2 (b^2 x + a^2 y) z (c^4 x^2 y^2 + 2 b^2 c^2 x^2 y z +
a^2 c^2 x y^2 z + b^4 x^2 z^2 + a^2 b^2 x y z^2 +
a^4 y^2 z^2) (c^4 x^2 y^2 + b^2 c^2 x^2 y z + 2 a^2 c^2 x y^2 z +
b^4 x^2 z^2 + a^2 b^2 x y z^2 + a^4 y^2 z^2)}

For 2. the perspector is

{a^2 (-c^16 x^8 y^8 - 6 c^14 x^7 y^7 (b^2 x + a^2 y) z -
c^12 x^6 y^6 (17 b^4 x^2 + 30 a^2 b^2 x y + 14 a^4 y^2) z^2 -
c^10 x^5 y^5 (30 b^6 x^3 + 69 a^2 b^4 x^2 y + 54 a^4 b^2 x y^2 +
14 a^6 y^3) z^3 -
3 b^2 c^8 x^5 y^4 (12 b^6 x^3 + 33 a^2 b^4 x^2 y +
31 a^4 b^2 x y^2 + 10 a^6 y^3) z^4 -
c^6 x^3 y^3 (30 b^10 x^5 + 99 a^2 b^8 x^4 y +
107 a^4 b^6 x^3 y^2 + 21 a^6 b^4 x^2 y^3 - 30 a^8 b^2 x y^4 -
14 a^10 y^5) z^5 -
c^4 x^2 y^2 (b^2 x + a^2 y)^3 (17 b^6 x^3 + 18 a^2 b^4 x^2 y -
12 a^4 b^2 x y^2 - 14 a^6 y^3) z^6 -
6 c^2 x y (b^2 x - a^2 y) (b^2 x + a^2 y)^6 z^7 - (b^2 x -
a^2 y) (b^2 x + a^2 y)^7 z^8),
b^2 (-c^16 x^8 y^8 - 6 c^14 x^7 y^7 (b^2 x + a^2 y) z -
c^12 x^6 y^6 (14 b^4 x^2 + 30 a^2 b^2 x y + 17 a^4 y^2) z^2 -
c^10 x^5 y^5 (14 b^6 x^3 + 54 a^2 b^4 x^2 y + 69 a^4 b^2 x y^2 +
30 a^6 y^3) z^3 -
3 a^2 c^8 x^4 y^5 (10 b^6 x^3 + 31 a^2 b^4 x^2 y +
33 a^4 b^2 x y^2 + 12 a^6 y^3) z^4 +
c^6 x^3 y^3 (14 b^10 x^5 + 30 a^2 b^8 x^4 y - 21 a^4 b^6 x^3 y^2 -
107 a^6 b^4 x^2 y^3 - 99 a^8 b^2 x y^4 - 30 a^10 y^5) z^5 +
c^4 x^2 y^2 (b^2 x + a^2 y)^3 (14 b^6 x^3 + 12 a^2 b^4 x^2 y -
18 a^4 b^2 x y^2 - 17 a^6 y^3) z^6 +
6 c^2 x y (b^2 x - a^2 y) (b^2 x + a^2 y)^6 z^7 + (b^2 x -
a^2 y) (b^2 x + a^2 y)^7 z^8),
c^2 (c^16 x^8 y^8 + 6 c^14 x^7 y^7 (b^2 x + a^2 y) z +
2 c^12 x^6 y^6 (7 b^4 x^2 + 15 a^2 b^2 x y + 7 a^4 y^2) z^2 +
2 c^10 x^5 y^5 (7 b^6 x^3 + 27 a^2 b^4 x^2 y + 27 a^4 b^2 x y^2 +
7 a^6 y^3) z^3 + 30 c^8 x^5 y^5 (a b^3 x + a^3 b y)^2 z^4 -
c^6 x^3 y^3 (14 b^10 x^5 + 30 a^2 b^8 x^4 y +
21 a^4 b^6 x^3 y^2 + 21 a^6 b^4 x^2 y^3 + 30 a^8 b^2 x y^4 +
14 a^10 y^5) z^5 -
c^4 x^2 y^2 (14 b^12 x^6 + 54 a^2 b^10 x^5 y +
93 a^4 b^8 x^4 y^2 + 107 a^6 b^6 x^3 y^3 +
93 a^8 b^4 x^2 y^4 + 54 a^10 b^2 x y^5 + 14 a^12 y^6) z^6 -
3 c^2 x y (b^2 x + a^2 y)^3 (2 b^8 x^4 + 4 a^2 b^6 x^3 y +
5 a^4 b^4 x^2 y^2 + 4 a^6 b^2 x y^3 + 2 a^8 y^4) z^7 - (b^2 x +
a^2 y)^4 (b^4 x^2 + a^2 b^2 x y + a^4 y^2)^2 z^8)}

For 3. the equation of the 12th curve is

c^12 x^6 y^6 + 5 b^2 c^10 x^6 y^5 z + 5 a^2 c^10 x^5 y^6 z +
11 b^4 c^8 x^6 y^4 z^2 + 23 a^2 b^2 c^8 x^5 y^5 z^2 +
11 a^4 c^8 x^4 y^6 z^2 + 14 b^6 c^6 x^6 y^3 z^3 +
44 a^2 b^4 c^6 x^5 y^4 z^3 + 44 a^4 b^2 c^6 x^4 y^5 z^3 +
14 a^6 c^6 x^3 y^6 z^3 + 11 b^8 c^4 x^6 y^2 z^4 +
44 a^2 b^6 c^4 x^5 y^3 z^4 + 65 a^4 b^4 c^4 x^4 y^4 z^4 +
44 a^6 b^2 c^4 x^3 y^5 z^4 + 11 a^8 c^4 x^2 y^6 z^4 +
5 b^10 c^2 x^6 y z^5 + 23 a^2 b^8 c^2 x^5 y^2 z^5 +
44 a^4 b^6 c^2 x^4 y^3 z^5 + 44 a^6 b^4 c^2 x^3 y^4 z^5 +
23 a^8 b^2 c^2 x^2 y^5 z^5 + 5 a^10 c^2 x y^6 z^5 + b^12 x^6 z^6 +
5 a^2 b^10 x^5 y z^6 + 11 a^4 b^8 x^4 y^2 z^6 +
14 a^6 b^6 x^3 y^3 z^6 + 11 a^8 b^4 x^2 y^4 z^6 +
5 a^10 b^2 x y^5 z^6 + a^12 y^6 z^6 = 0.

Francisco Javier García Capitán
27 January 2012

----------------------------------------------------

Variation:

Let ABC be triangle, P a point, A1B1C1 the pedal (instead of cevian) triangle of P and A2B2C2 the circumcevian triangle of P. etc

APH


Τετάρτη, 25 Ιανουαρίου 2012

LOCUS


Let ABC be a triangle, A'B'C' the cevian triangle of G, A"B"C" the circumcevian triangle of G with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of GB"C",GC"A",GA"B", resp.


The triangles ABC, O1O2O3 are perspective.

Perspector?

APH, 25 January 2012

-------------------------------------------------

Generalization:

Let ABC be a triangle, A'B'C' the cevian triangle of P, A"B"C" the circumcevian triangle of P with respect the triangle A'B'C' and O1,O2,O3 the circumcenters of PB"C", PC"A", PA"B", resp.
The triangles ABC, O1O2O3 are perspective gives as locus the Yiu quintic and an octic through G and H.

The perspector for H is X381 and that for G is the isotomic conjugate of the point {3 a^4 - 4 a^2 b^2 + b^4 - 4 a^2 c^2 - 6 b^2 c^2 + c^4,
a^4 - 4 a^2 b^2 + 3 b^4 - 6 a^2 c^2 - 4 b^2 c^2 + c^4,
a^4 - 6 a^2 b^2 + b^4 - 4 a^2 c^2 - 4 b^2 c^2 + 3 c^4}, not in ETC.

Francisco Javier García Capitán
26 January 2012

ZIG ZAG LOCUS


Let ABC be a triangle, P a point (not on the circumcircle), A1B1C1 the circumcevian triangle of P, A2B2C2 the circumcevian triangle of H with respect the triangle A1B1C1 and A3B3C3 the circumcevian triangle of O with respect the triangle A2B2C2 (antipodal triangle of A2B2C2).


Which is the locus of P such that the triangles ABC, A3B3C3 are perspective? And the locus of the perspectors?
Generalization: Replace H with a point Q.

APH, 25 January 2012

---------------------------------------------------------

The locus is the Euler line.
For an arbitrary Q, instead of H, the locus is the line OQ, and the locus of perspectors P' is the same line, so we have a map P -> P', although the formula is not simple.

I found the following formula showing the relationship between some P on line OQ such that OP:PQ = k, and the perspector P', also on line OQ such that OP':P'Q = k': k' = R^2 / (k OQ^2 - (k+2) R^2)

Francisco Javier García Capitán
25 January 2012

---------------------------------------------------------

Variation:

A2B2C2 = the circumcevian triangle of P* (= the isogonal or isotomic conjugate of P, instead of H) with respect the triangle A1B1C1

APH

Δευτέρα, 23 Ιανουαρίου 2012

SEGOVIA POINT Continued


The antipodal triangle A'B'C' of O (= circumcevian of O) is the reflection of ABC in O. The triangle A2B2C2 is the reflection of the orthic A1B1C1 in N. Consider now the triangle A'2B'2C'2 = the reflection of A2B2C2 in O.


The triangles A'B'C', A2B2C2 are perspective at SEGOVIA Point of ABC.

Similarly the triangles ABC, A'2B'2C'2 are perspective at the SEGOVIA point of A'B'C'.

Denote O1O2O3 = the pedal triangle of O (=medial triangle).

We will work in an acute triangle ABC (similarly if ABC is not acute. Simply we have to change some signs).

ABC, A'2B'2C'2 are perspective <==>

[cotB + cot(O1BA'2)] / [cotC + cot(O1CA'2] * Cyclically = 1

We have:

cot(O1BA'2) = O1B / O1A'2 = (BC/2)/[OA'2 - OO1] = (BC/2)/[HA1 - OO1] =

= sinA /(2cosBcosC - cosA)

Therefore:

[cotB + cot(O1BA'2)] = (cosB/sinB) - [sinA /(2cosBcosC - cosA)] =

cosC[1 + 2cos^2B] / sinB(2cosBcosC - cosA)

and

[cotB + cot(O1BA'2)] / [cotC + cot(O1CA'2] = (cosC/cosB)*(sinC/sinB)*[(1+2cos^2B)/(1+2cos^2C)] (*)

and similarly the other two ratia.

Multiplicating them we get 1, therefore the triangles are perspective.

As for the coordinates of the perspector:

We get it in barycentrics from (*) and are:

(cosA * sinA * (1/(1+2cos^2A)) ::)

These are the barycentrics of the Segovia Point of the circumcevian triangle of O (if I did not make some computational error!)

Κυριακή, 22 Ιανουαρίου 2012

BROCARD PRIZE



Let ABC be a triangle and Ka, Kb, Kc the Brocard axes of
the triangles GBC, GCA, GAB, resp.

Let A'B'C' be a triangle homothetic and sharing the same
centroid G with ABC.

Conjecture:

The reflections La,Lb,Lc of Ka,Kb,Kc in the sidelines B'C', C'A',
A'B' of A'B'C' resp. are concurrent.

See: ANOPOLIS list, Message 137

The first who will send a solution (synthetic or not) to list HYACINTHOS will win the book: F.G.-M.: Exercices d' Algebre (1198 pages)



Good Luck!

Τετάρτη, 18 Ιανουαρίου 2012

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!

Τρίτη, 17 Ιανουαρίου 2012

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

---------------------------------------------------------

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.

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

---------------------------------------------

This is the midpoint of DH.

Francisco Javier García Capitán
17 January 2012

Δευτέρα, 16 Ιανουαρίου 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

Σάββατο, 14 Ιανουαρίου 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

---------------------------------------------

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)
--------------------------------------------

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

--------------------------------------------

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

--------------------------------------------

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

--------------------------------------------

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

------------------------------------------------------------

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

Παρασκευή, 13 Ιανουαρίου 2012

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

 

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