[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.
D = the Poncelet point of ABCN ( = X(137)
A'B'C' = the pedal triangle of D wrt triangle NaNbNc
ABC, A'B'C' are homothetic.
Loci:
1. Let ABC be a triangle and P a point..
Denote:
Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.
A'B'C' = the pedal triangle of P wrt triangle NaNbNc
Which is the locus of P such that ABC, A'B'C' are:
1. perspective ?
2. orthologic ?
2. Let ABC be a triangle and P a point.
Denote:
Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.
D = the Poncelet point of ABCP
A'B'C' = the pedal triangle of D wrt triangle NaNbNc
Which is the locus of P such that ABC, A'B'C' are:
1. perspective?
2. orthologic?
--------------------------------------------------------------------------------------------
[Ercole Suppa]
Answers to first two questions (1.1) and (2.2)
(1.1)
*** Locus of point P such that ABC, A'B'C' are perspective = {cubic q3 with equation very complicated}
-- q3 pass through X(i) for these i : {137,1263}
-- pares {P=X(i),Q=X(j)} for these {i,j}: {1263,1263}
-- some points Q1(X(P):
P1=Q1(X(137)) = X(4)X(12026) ∩ X(5)X(128)
= (-(b^2-c^2)^2+a^2 (b^2+c^2)) (a^12-4 a^10 (b^2+c^2)+(b^2-c^2)^4 (3 b^4-2 b^2 c^2+3 c^4)+a^8 (9 b^4+4 b^2 c^2+9 c^4)-4 a^2 (b^2-c^2)^2 (3 b^6-b^4 c^2-b^2 c^4+3 c^6)-4 a^6 (4 b^6-b^4 c^2-b^2 c^4+4 c^6)+a^4 (19 b^8-18 b^6 c^2+7 b^4 c^4-18 b^2 c^6+19 c^8)) : : (barys)
= X[4]+2*X[12026], 5*X[5]-2*X[128], 2*X[546]+X[1141], X[930]-4*X[3628], 5*X[1656]-2*X[6592], 7*X[3090]-X[13512], X[6343]-4*X[8254]
= lies on these lines : {4,12026},{5,128},{381,9143},{546,1141},{930,3628},{1656,6592},{3090,13512},{3327,10593},{3459,14143},{6343,8254},{7159,10592},{7741,14101},{10126,20413},{13406,15367},{14652,18378},{15307,22051}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {5,137,1263},{5,1263,14072},{1656,11671,6592}
= (6-8-13) search numbers [-1.89954180405816773, -1.77162773310843386, 5.74388759131706457]
--------------------------------------------
(1.2)
*** Locus of point P such that ABC, A'B'C' are orthologic is:
Γ={L=line through X(137),X(5501),X(8254),X(14051),X(14071)} U {c1=circunference with center X(5501) and radius r1}, where
r1=((2 p^4-12 p^2 r^2+2 r^4-16 p^2 r R+16 r^3 R-8 p^2 R^2+40 r^2 R^2+32 r R^3+9 R^4) (32 p^4 r^2-32 p^2 r^4-128 p^2 r^3 R-112 p^2 r^2 R^2+2 p^2 R^4-2 r^2 R^4-8 r R^5+R^6))/(64 (p^4+2 p^2 r^2+r^4-8 p^2 r R+8 r^3 R-6 p^2 R^2+22 r^2 R^2+24 r R^3+8 R^4)^2)
and p,r,R are the semiperimeter, inradius, sircumradius of ABC resp.
*** c1: pass through points X(i) for these i: {10095,20327,20414}
*** The locus of the point of concurrence Q=Q(P):
-- if P ∈ L then Q(P) ∈ rectangular circum-hyperbola h2 with center X(137), perspector X(12077), through X(i) for these i: {4,5,53,311,327,1141,1263,1487,2165,2980,3459,3613,8797,8800,10412,11082,11087,11816,13450,14225,15619,16837,17500,17507,17703,19712,19713,21011,22261,22335}
-- if P ∈ c1 then Q(P) ∈ Linf
*** pairs (P,Q(P))
-- pairs (P ∈ L, Q(P) ∈ h2): {137,4},{5501,1263},{8254},{14071,3459}
-- pairs (P ∈ c1, Q(P)) : none
*** some points Q=Q2(P) :
P2=Q2(X(14051)) = ISOGONAL CONJUGATE OF X(6150)
= (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-4 a^8 c^2+5 a^6 b^2 c^2-2 a^4 b^4 c^2+5 a^2 b^6 c^2-4 b^8 c^2+8 a^6 c^4+a^4 b^2 c^4+a^2 b^4 c^4+8 b^6 c^4-10 a^4 c^6-10 a^2 b^2 c^6-10 b^4 c^6+7 a^2 c^8+7 b^2 c^8-2 c^10) (a^10-4 a^8 b^2+8 a^6 b^4-10 a^4 b^6+7 a^2 b^8-2 b^10-3 a^8 c^2+5 a^6 b^2 c^2+a^4 b^4 c^2-10 a^2 b^6 c^2+7 b^8 c^2+2 a^6 c^4-2 a^4 b^2 c^4+a^2 b^4 c^4-10 b^6 c^4+2 a^4 c^6+5 a^2 b^2 c^6+8 b^4 c^6-3 a^2 c^8-4 b^2 c^8+c^10) : : (barys)
= lies on the curve Q106 and these lines: {5,930},{137,252},{1263,19552},{10126,20413},{15643,21394},{20030,22335}
= isogonal conjugate of X(6150)
= reflection of X(i) in X(j) for these {i,j}: {930,21975},{3459,137}
= (6-8-13) search numbers [-5.94394753050580450, -5.37413518099615100, 10.1045800059074703]
P3=Q2(X(10095)) = X(30)X(511) ∩ X(128)X(8562)
= (b-c) (b+c) (-a^6+2 a^4 b^2-a^2 b^4-a^2 b^3 c+b^5 c+2 a^4 c^2+a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3-a^2 c^4+b c^5) (a^6-2 a^4 b^2+a^2 b^4-a^2 b^3 c+b^5 c-2 a^4 c^2-a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3+a^2 c^4+b c^5) : : (barys)
= lies on these lines: {30,511},{128,8562},{137,8901},{1141,10412},{6132,14769},{14225,15619}
= (6-8-13) search numbers [1.09518203377538903, -7.03768486777356179, 4.36677473902382476]
P4=Q2(X(20327)) = ISOGONAL CONJUGATE OF X(15907)
= (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^12-4 a^10 b^2+6 a^8 b^4-4 a^6 b^6+a^4 b^8-4 a^10 c^2+10 a^8 b^2 c^2-8 a^6 b^4 c^2+3 a^4 b^6 c^2-2 a^2 b^8 c^2+b^10 c^2+6 a^8 c^4-8 a^6 b^2 c^4+a^4 b^4 c^4+2 a^2 b^6 c^4-4 b^8 c^4-4 a^6 c^6+3 a^4 b^2 c^6+2 a^2 b^4 c^6+6 b^6 c^6+a^4 c^8-2 a^2 b^2 c^8-4 b^4 c^8+b^2 c^10) : : (barys)
= lies on these lines: {1,3327},{3,252},{4,11671},{5,128},{12,14101},{26,15959},{30,511},{54,14071},{140,6592},{156,6069},{1157,19553},{5576,14769},{5889,13505},{7387,15960},{7512,14652},{7568,13467},{10205,14143},{11412,13504},{13160,15367},{14118,14674},{14865,14978},{15425,15957}
= isogonal conjugate of X(15907)
= complement of isogonal conjugate of X(15907)
= anticomplement of isogonal conjugate of X(15907)
= antigonal image of isogonal conjugate of X(15907)
= (6-8-13) search numbers [1.54025249931274447, 0.912892302702155597, -1.34288813232275901]
Best regards
Ercole Suppa
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου