[Antreas P. Hatzipolakis]:
Let ABC be a triangle and L a line.
Denote:
(Oa), (Ob), (Oc) = the reflections of the circumcircle (O) in BC, CA, AB, resp
The parallel through A to L intersects (Ob), (Oc) at Ba, Ca, resp.
The parallel through B to L intersects (Oc), (Oa) at Cb, Ab, resp.
The parallel through C to L intersects (Oa), (Ob) at Ac, Bc, resp.
A*B*C* = the triangle bounded by the lines AbAc, BcBa, CaCb
ABC, A*B*C* are circumparallelogic
ie the parallelogic center (ABC, A*B*C*) lies on the curcumcircle of ABC.
(that is, the parallels through A, B, C, to B*C*, C*A*, A*B*, resp. concur on the circumcircle of ABC)
the parallelogic center (A*B*C*, ABC) lies on the circumcircle of A*B*C*.
(that is, the parallels through A*, B*, C* to BC, CA, AB, resp. concur on the circumcircle of A*B*C*)
Which are the parallelogic centers?
Special L = Euler line, Brocard axis, OI line .......
[Ercole Suppa]:
Hi Antreas,
Let L: u x + v y + w z = 0 be the barycentric equation of line L, let Z1 = circumparallelogic center (ABC,A*B*C*) and Z2 = circumparallelogic center (A*B*C*,ABC)
We have:
Z1= (a^2 (u - w)^2 - b^2 (v - w)^2) (-a^2 (u - v)^2 + c^2 (v - w)^2) : : (barys)
Z2= 2 (v-w)^3 (b^4 (u-v)+c^4 (-u+w))-a^2 (u-v) (u-w) (v-w) (b^2 (3 u+v-4 w)-c^2 (3 u-4 v+w))+a^4 (u-v) (u-w) (5 u^2+2 v^2+v w+2 w^2-5 u (v+w)) :: (barys)
Circumparallelogic centers for some special lines:
*** L = Euler line
Z1 = X(476)
Z2 = X(1553)X(14094) ∩ X(5627)X(14611)
= 3 a^16-10 a^14 b^2+7 a^12 b^4+12 a^10 b^6-25 a^8 b^8+22 a^6 b^10-15 a^4 b^12+8 a^2 b^14-2 b^16-10 a^14 c^2+36 a^12 b^2 c^2-42 a^10 b^4 c^2+17 a^8 b^6 c^2-15 a^6 b^8 c^2+33 a^4 b^10 c^2-25 a^2 b^12 c^2+6 b^14 c^2+7 a^12 c^4-42 a^10 b^2 c^4+51 a^8 b^4 c^4-11 a^6 b^6 c^4-36 a^4 b^8 c^4+27 a^2 b^10 c^4+4 b^12 c^4+12 a^10 c^6+17 a^8 b^2 c^6-11 a^6 b^4 c^6+36 a^4 b^6 c^6-10 a^2 b^8 c^6-38 b^10 c^6-25 a^8 c^8-15 a^6 b^2 c^8-36 a^4 b^4 c^8-10 a^2 b^6 c^8+60 b^8 c^8+22 a^6 c^10+33 a^4 b^2 c^10+27 a^2 b^4 c^10-38 b^6 c^10-15 a^4 c^12-25 a^2 b^2 c^12+4 b^4 c^12+8 a^2 c^14+6 b^2 c^14-2 c^16 : : (barys)
= 2 S^4 + (-63 R^2 SB-63 R^2 SC-6 SB SC+33 R^2 SW+14 SB SW+14 SC SW-8 SW^2)S^2 -162 R^4 SB SC+99 R^2 SB SC SW-12 SB SC SW^2 : : (barys)
= 4*X[1553]-3*X[14094], 3*X[5627]-2*X[14611], 4*X[6070]-3*X[15035], 2*X[14480]-3*X[14644]
= lies on these lines: {1553,14094}, {5627,14611}, {6070,15035},{14480,14644}
= (6-9-13) search numbers [-2.37936584815301071, 3.31210754266471833, 2.44583580536404059]
*** L = Brocard axis
Z1 = X(805)
Z2 = (name pending)
= a^2 (2 a^12 b^4-6 a^10 b^6+8 a^8 b^8-8 a^6 b^10+6 a^4 b^12-2 a^2 b^14-a^12 b^2 c^2-4 a^10 b^4 c^2+5 a^8 b^6 c^2-3 a^6 b^8 c^2-2 a^4 b^10 c^2+5 a^2 b^12 c^2+2 a^12 c^4-4 a^10 b^2 c^4+24 a^8 b^4 c^4-19 a^6 b^6 c^4+15 a^4 b^8 c^4-15 a^2 b^10 c^4+3 b^12 c^4-6 a^10 c^6+5 a^8 b^2 c^6-19 a^6 b^4 c^6-3 a^4 b^6 c^6+8 a^2 b^8 c^6-11 b^10 c^6+8 a^8 c^8-3 a^6 b^2 c^8+15 a^4 b^4 c^8+8 a^2 b^6 c^8+16 b^8 c^8-8 a^6 c^10-2 a^4 b^2 c^10-15 a^2 b^4 c^10-11 b^6 c^10+6 a^4 c^12+5 a^2 b^2 c^12+3 b^4 c^12-2 a^2 c^14) : : (barys)
= (3 R^2+2 SB+2 SC-2 SW)S^6 +(-21 R^2 SB SW-21 R^2 SC SW+2 SB SC SW+5 R^2 SW^2)S^4 + (21 R^2 SB SC SW^2+15 R^2 SB SW^3+15 R^2 SC SW^3-10 R^2 SW^4-2 SB SW^4-2 SC SW^4+2 SW^5)S^2 + R^2 SB SC SW^4-2 SB SC SW^5 : : (barys)
= 3*X[6054]-4*X[13137]
= lies on this line: {6054,13137}
= (6-9-13) search numbers [-18.4251556098146610, 26.6750112610137398, -6.32273303272605877]
*** L = OI line
Z1 = X(901)
Z2 = a^2 (2 a^9 b^2-2 a^8 b^3-8 a^7 b^4+8 a^6 b^5+12 a^5 b^6-12 a^4 b^7-8 a^3 b^8+8 a^2 b^9+2 a b^10-2 b^11-a^9 b c-4 a^8 b^2 c+18 a^7 b^3 c+6 a^6 b^4 c-48 a^5 b^5 c+6 a^4 b^6 c+46 a^3 b^7 c-14 a^2 b^8 c-15 a b^9 c+6 b^10 c+2 a^9 c^2-4 a^8 b c^2-10 a^7 b^2 c^2-4 a^6 b^3 c^2+33 a^5 b^4 c^2+21 a^4 b^5 c^2-44 a^3 b^6 c^2-14 a^2 b^7 c^2+19 a b^8 c^2+b^9 c^2-2 a^8 c^3+18 a^7 b c^3-4 a^6 b^2 c^3-39 a^5 b^3 c^3+a^4 b^4 c^3+10 a^3 b^5 c^3+18 a^2 b^6 c^3+11 a b^7 c^3-13 b^8 c^3-8 a^7 c^4+6 a^6 b c^4+33 a^5 b^2 c^4+a^4 b^3 c^4-16 a^3 b^4 c^4+2 a^2 b^5 c^4-21 a b^6 c^4+3 b^7 c^4+8 a^6 c^5-48 a^5 b c^5+21 a^4 b^2 c^5+10 a^3 b^3 c^5+2 a^2 b^4 c^5+8 a b^5 c^5+5 b^6 c^5+12 a^5 c^6+6 a^4 b c^6-44 a^3 b^2 c^6+18 a^2 b^3 c^6-21 a b^4 c^6+5 b^5 c^6-12 a^4 c^7+46 a^3 b c^7-14 a^2 b^2 c^7+11 a b^3 c^7+3 b^4 c^7-8 a^3 c^8-14 a^2 b c^8+19 a b^2 c^8-13 b^3 c^8+8 a^2 c^9-15 a b c^9+b^2 c^9+2 a c^10+6 b c^10-2 c^11) : : (barys)
= (6-9-13) search numbers [-15.0176195790816926, 19.6791605798079685, -3.05216072915266402]
*** L = DeLongchamps line or L = orthic axis
(notice that DeLongchamps line and orthic axis are parallel lines)
Z1= X(476)
Z2 = REFLECTION OF X(14731) IN X(24981)
= 5 a^12-10 a^10 b^2+4 a^8 b^4+a^4 b^8+2 a^2 b^10-2 b^12-10 a^10 c^2+22 a^8 b^2 c^2-10 a^6 b^4 c^2-a^4 b^6 c^2-7 a^2 b^8 c^2+6 b^10 c^2+4 a^8 c^4-10 a^6 b^2 c^4+5 a^4 b^4 c^4+5 a^2 b^6 c^4-6 b^8 c^4-a^4 b^2 c^6+5 a^2 b^4 c^6+4 b^6 c^6+a^4 c^8-7 a^2 b^2 c^8-6 b^4 c^8+2 a^2 c^10+6 b^2 c^10-2 c^12 : : (barys)
= 2 S^4+ (-54 R^4+27 R^2 SB+27 R^2 SC-6 SB SC+27 R^2 SW-6 SB SW-6 SC SW-4 SW^2)S^2 -63 R^2 SB SC SW+16 SB SC SW^2 : : (barys)
= 5*X[110]-4*X[3258], 4*X[7471]-3*X[9140], 16*X[12068]-15*X[15059], X[14731]-2*X[24981], 5*X[15034]-4*X[16340]
= lies on these lines: {30,14094}, {110,3258}, {7471,9140}, {12068,15059}, {14731,24981}, {15034,16340}
= reflection of X(14731) in X(24981)
= (6-9-13) search numbers [-11.2914358286080317, -14.9403328671974973, 19.1954037724017380]
*** L = antiorthic axis
Z1 = X(901)
Z2 = (name pending)
= a^2 (2 a^5 b^2-2 a^4 b^3-4 a^3 b^4+4 a^2 b^5+2 a b^6-2 b^7+a^5 b c-8 a^4 b^2 c+12 a^3 b^3 c+2 a^2 b^4 c-13 a b^5 c+6 b^6 c+2 a^5 c^2-8 a^4 b c^2+14 a^3 b^2 c^2-16 a^2 b^3 c^2+11 a b^4 c^2-3 b^5 c^2-2 a^4 c^3+12 a^3 b c^3-16 a^2 b^2 c^3+5 a b^3 c^3-b^4 c^3-4 a^3 c^4+2 a^2 b c^4+11 a b^2 c^4-b^3 c^4+4 a^2 c^5-13 a b c^5-3 b^2 c^5+2 a c^6+6 b c^6-2 c^7) : : (barys)
= 5*X[100]-4*X[3025]
= lies on this line: {100,3025}
= (6-9-13) search numbers [-0.345233799974865728, -16.0565402275804837, 14.9160694702208058]
*** L = Lemoine axis
Z1 = X(805)
Z2 = (name pending)
= a^2 (2 a^8 b^4-6 a^6 b^6+6 a^4 b^8-2 a^2 b^10+a^8 b^2 c^2-4 a^6 b^4 c^2+2 a^4 b^6 c^2-a^2 b^8 c^2+2 a^8 c^4-4 a^6 b^2 c^4+14 a^4 b^4 c^4-7 a^2 b^6 c^4+3 b^8 c^4-6 a^6 c^6+2 a^4 b^2 c^6-7 a^2 b^4 c^6-b^6 c^6+6 a^4 c^8-a^2 b^2 c^8+3 b^4 c^8-2 a^2 c^10) : : (barys)
= 3*X[671]-4*X[12833]
= lies on this line: {671,12833}
= (6-9-13) search numbers [5.75572955191116459, -15.3789065585625194, 11.6311092292609704]
Best regards,
Ercole Suppa
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