HYACINTHOS 29114

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.

Denote:

Na, Nb, Nc  = the NPC centers of PBC, PCA, PAB, resp.

D = the Poncelet poin of ABCP

Da, Db, Dc = the orthogonal projections of D on AP, BP, CP, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in DDa, DDb, DDc, resp.

The parallels to DN1, DN2, DN3 through A', B', C', resp. concur on the pedal circle of P.

Point of concurrence?  

 

[Peter Moses]:


Concurrence:
[....]
On the pedal circle of P.

Examples:
P = X(1) -> X(13756).
P = X(3) -> X(25641).

[APH]:

For P = X(4) = H:

In this case Na, Nb, Nc coincide with N and the Poncelet point of ABCH = D is any point on the pedal circle of H = NPC.

So we have this problem:

Let ABC be a triangle and A'B'C' the pedal triangle of H (orthic triangle)

Denote:

D = a point on the NPC

Da, Db, Dc = the orthogonal projections of D on AH, BH, CH, resp.

N1, N2, N3 = the reflections of N in DDa, DDb, DDc, resp.

The parallels to DN1, DN2, DN3 through A', B', C', resp. concur on the NPC.

Which is the point of concurrence for some D's ?
(D = X(11), X(125), X(137) .......)


[Peter Moses]:

Hi Antreas,

> Which is the point of concurrence for some D's ?
> (D = X(11), X(125), X(137) .......)

X(11)  ->  X(31841)
X(113)  ->  X(25641)
X(119)  ->  X(31841)
X(125)  ->  X(25641)
X(126)  ->  X(6092)
X(1312)  ->  X(113)
X(1313)  ->  X(113)
X(2039)  ->  X(114)
X(2040)  ->  X(114)
X(5512)  ->  X(6092)
X(13870)  ->  X(16188)
X(13872)  ->  X(31842)

Let Q = antipode of P{p,q,r} in the nine point circle, then
P & Q  ->  ((2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) p+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) q+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) r)((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) p+(a^4 b^2-2 a^2 b^4+b^6-a^4 c^2+2 a^2 b^2 c^2-3 b^4 c^2+2 a^2 c^4+3 b^2 c^4-c^6) q-(a^4 b^2-2 a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-3 b^4 c^2+2 a^2 c^4+3 b^2 c^4-c^6) r) : :

= X(4)-Ceva conjugate of (line(X(5), P) /\ {1,1,1}).
= complementary conjugate of (line(X(5), P) /\ {1,1,1}).
= complement of the (isogonal conjugate of (line(X(5), P) /\ {1,1,1})).
= line[Q, complementary conjugate P] /\ line[P, complementary conjugate Q].

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

X(114) & X(115) ->  COMPLEMENT OF X(2698)

 

= a^2*(a^6*b^2 - a^4*b^4 + a^6*c^2 - 2*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6)*(a^4*b^4 - 2*a^2*b^6 + b^8 - 2*b^6*c^2 + a^4*c^4 + 4*b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) : :

= X[14510] - 3 X[14651],3 X[14639] - X[31513]

= lies on the nine-point circle and these lines: {2, 2698}, {3, 22103}, {4, 805}, {5, 2679}, {114, 512}, {115, 511}, {125, 21531}, {127, 5031}, {136, 297}, {137, 11675}, {626, 31848}, {1567, 2782}, {2080, 8623}, {3258, 5650}, {5099, 24206}, {5446, 16979}, {6033, 9467}, {12833, 13137}, {14510, 14651}, {14639, 31513}
= complement of X(2698)
= midpoint of X(i) and X(j) for these {i,j}: {4, 805}, {6071, 6072}, {12833, 13137}
= reflection of X(i) in X(j) for these {i,j}: {3, 22103}, {2679, 5}, {16979, 5446}
= complement of the isogonal of X(2782)
= orthic isogonal conjugate of X(2782)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 2782}, {2782, 10}, {16068, 18904}
= X(4)-Ceva conjugate of X(2782)

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

X(116) & X(118) -> COMPLEMENT OF X(2724)

 

= (a^4*b^2 - 2*a^3*b^3 + 2*a*b^5 - b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 + 4*b^3*c^3 - b^2*c^4 + 2*a*c^5 - c^6)*(2*a^6 - 2*a^5*b + a^4*b^2 - 2*a^3*b^3 + b^6 - 2*a^5*c + 2*a^3*b^2*c + 2*a^2*b^3*c - 2*b^5*c + a^4*c^2 + 2*a^3*b*c^2 - 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 - b^2*c^4 - 2*b*c^5 + c^6) : :

= 5 X[3091] - X[14732]

 
= lies on the nine-point circle and these lines: {2, 2724}, {4, 927}, {5, 1566}, {11, 241}, {116, 516}, {118, 514}, {123, 20544}, {125, 857}, {1517, 2808}, {3091, 14732}, {3259, 9779}, {5519, 18343}, {10739, 14942}
= complement of X(2724)
= midpoint of X(i) and X(j) for these {i,j}: {4, 927}, {6074, 14505}
= reflection of X(1566) in X(5)
= complement of the isogonal of X(2808)
= orthic isogonal conjugate of X(2808)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 2808}, {2808, 10}, {23694, 518}
= X(4)-Ceva conjugate of X(2808)
---------------------------------------------------

X(126) & X(5512) -> COMPLEMENT OF X(6093)

 

= (a^6*b^2 + a^4*b^4 - a^2*b^6 - b^8 + a^6*c^2 - 14*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 7*b^6*c^2 + a^4*c^4 + 10*a^2*b^2*c^4 - 20*b^4*c^4 - a^2*c^6 + 7*b^2*c^6 - c^8)*(2*a^8 - 6*a^6*b^2 + 19*a^4*b^4 - 8*a^2*b^6 + b^8 - 6*a^6*c^2 - 20*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 19*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - 8*a^2*c^6 + c^8) : :

= lies on the nine-point circle and these lines: {2, 6093}, {4, 6082}, {5, 31654}, {125, 8355}, {126, 1499}, {524, 5512}, {599, 5099}, {5139, 16183}, {6076, 6077}, {20388, 20389}
= complement of X(6093)
= midpoint of X(i) and X(j) for these {i,j}: {4, 6082}, {6076, 6077}
= reflection of X(31654) in X(5)

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

X(128) & X(137) -> COMPLEMENT OF X(15907)

 

= a^2*(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)*(a^12*b^4 - 6*a^10*b^6 + 15*a^8*b^8 - 20*a^6*b^10 + 15*a^4*b^12 - 6*a^2*b^14 + b^16 - 2*a^10*b^4*c^2 + 4*a^8*b^6*c^2 + 4*a^6*b^8*c^2 - 16*a^4*b^10*c^2 + 14*a^2*b^12*c^2 - 4*b^14*c^2 + a^12*c^4 - 2*a^10*b^2*c^4 + 6*a^8*b^4*c^4 - 8*a^6*b^6*c^4 + 10*a^4*b^8*c^4 - 16*a^2*b^10*c^4 + 9*b^12*c^4 - 6*a^10*c^6 + 4*a^8*b^2*c^6 - 8*a^6*b^4*c^6 + 8*a^2*b^8*c^6 - 16*b^10*c^6 + 15*a^8*c^8 + 4*a^6*b^2*c^8 + 10*a^4*b^4*c^8 + 8*a^2*b^6*c^8 + 20*b^8*c^8 - 20*a^6*c^10 - 16*a^4*b^2*c^10 - 16*a^2*b^4*c^10 - 16*b^6*c^10 + 15*a^4*c^12 + 14*a^2*b^2*c^12 + 9*b^4*c^12 - 6*a^2*c^14 - 4*b^2*c^14 + c^16) : :

= lies on the nine-point circle and these lines: {2, 15907}, {125, 16336}, {128, 1510}, {136, 14106}, {137, 1154}, {3258, 32142}, {11583, 11792}
= complement of X(15907)
= complement of the isogonal of X(25150)
= orthic isogonal conjugate of X(25150)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 25150}, {25150, 10}
= X(4)-Ceva conjugate of X(25150)

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

Best regards,

Peter Moses.