HYACINTHOS 29086

[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 pouint of ABCP

DNa, DNb, DNc intersect the pedal circle of P ( = circumcircle of A'B'C') at A*, B*, C*, resp.

The lines A'A*, B'B*, C'C* bound a triangle A"B"C".

A"B"C", NaNbNc are parallelogic.

Parallelogic centers in terms of P?


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[Ercole Suppa]:

Let P(u:v:w) be the barycentric coordinates of P. 

A"B"C", NaNbNc are parallelogic and the parallelogic centers are:

Z1(P) = Z(A"B"C", NaNbNc) = -b^2 c^2 (b^2-c^2)^2 u^4 (-c^2 v+b^2 (u+v)) (v+w)^3 (c^2 v+b^2 w) (b^2 w-c^2 (u+w))+a^12 v^2 (u+v) w^2 (u+w) (c^2 v (u+2 v) (u+w)^2+b^2 (u+v)^2 w (u+2 w))-a^10 v w (b^4 (u+v)^3 w^2 (5 u^2 v+2 w^2 (2 v+w)+2 u w (5 v+w))+c^4 v^2 (u+w)^3 (5 u^2 w+2 v^2 (v+2 w)+2 u v (v+5 w))+2 b^2 c^2 v (u+v) w (u+w) (-u^4+5 u v w (v+w)+v w (v^2+w^2)+u^2 (4 v^2+v w+4 w^2)))+a^8 (b^6 (u+v)^3 w^3 (2 v w^2 (v+w)+u^2 (10 v^2+2 v w-w^2)+u w (12 v^2+6 v w-w^2))+c^6 v^3 (u+w)^3 (2 v^2 w (v+w)+u^2 (-v^2+2 v w+10 w^2)+u v (-v^2+6 v w+12 w^2))+b^4 c^2 v (u+v) w^2 (-8 u^5 v+2 v^2 w^2 (v^2-v w-2 w^2)-u^4 (11 v^2+5 v w+3 w^2)+u v w (5 v^3-6 v^2 w+6 v w^2+2 w^3)+u^3 (12 v^3-13 v^2 w+6 v w^2+4 w^3)+u^2 w (12 v^3+12 v^2 w+7 w^3))-b^2 c^4 v^2 w (u+w) (8 u^5 w+2 v^2 w^2 (2 v^2+v w-w^2)+u^4 (3 v^2+5 v w+11 w^2)-u v w (2 v^3+6 v^2 w-6 v w^2+5 w^3)-u^3 (4 v^3+6 v^2 w-13 v w^2+12 w^3)-u^2 v (7 v^3+12 v w^2+12 w^3)))-a^2 (b^2-c^2) u^2 (v+w) (b^10 (u+v)^3 w^3 (v+w)-c^10 v^3 (u+w)^3 (v+w)-b^8 c^2 (u+v) w^2 (5 v^2 w (v+w)+u^3 (2 v+w)+u^2 (8 v^2+7 v w-w^2)+u w (5 v^2+6 v w+w^2))+b^2 c^8 v^2 (u+w) (5 v w^2 (v+w)+u^3 (v+2 w)+u v (v^2+6 v w+5 w^2)+u^2 (-v^2+7 v w+8 w^2))-b^4 c^6 v (10 v^2 w^3 (v+w)+u^4 (-3 v^2+v w+3 w^2)+u^3 (-2 v^3+4 v^2 w+6 v w^2-3 w^3)+u^2 (v^4-3 v^3 w+16 v^2 w^2+15 v w^3-5 w^4)+u w (2 v^4+13 v^3 w+15 v^2 w^2+5 v w^3+w^4))+b^6 c^4 w (10 v^3 w^2 (v+w)+u^4 (3 v^2+v w-3 w^2)+u^3 (-3 v^3+6 v^2 w+4 v w^2-2 w^3)+u^2 (-5 v^4+15 v^3 w+16 v^2 w^2-3 v w^3+w^4)+u v (v^4+5 v^3 w+15 v^2 w^2+13 v w^3+2 w^4)))+a^4 u (b^10 (u+v)^3 w^3 (v+w) (w (v+w)+u (5 v+w))+c^10 v^3 (u+w)^3 (v+w) (v (v+w)+u (v+5 w))-b^2 c^8 v^2 (u+w) (v (4 v-w) w^2 (v+w)^2+2 u^4 w (3 v+4 w)+u v w (3 v^3+4 v^2 w+19 v w^2+18 w^3)+u^3 (v^3+4 v^2 w+24 v w^2+21 w^3)-2 u^2 (v^4-3 v^3 w-16 v^2 w^2-11 v w^3+w^4))-b^8 c^2 (u+v) w^2 (-v^2 (v-4 w) w (v+w)^2+2 u^4 v (4 v+3 w)+u^3 (21 v^3+24 v^2 w+4 v w^2+w^3)+u v w (18 v^3+19 v^2 w+4 v w^2+3 w^3)-2 u^2 (v^4-11 v^3 w-16 v^2 w^2-3 v w^3+w^4))+b^6 c^4 w (-2 v^3 (2 v-3 w) w^2 (v+w)^2+u^4 v (-9 v^3+10 v^2 w+8 v w^2-5 w^3)+u^5 (3 v^3-3 v^2 w-5 v w^2+3 w^3)+u v^2 w (-v^4+22 v^3 w+12 v^2 w^2-6 v w^3+5 w^4)+u^2 v (5 v^5+4 v^4 w+26 v^3 w^2+29 v^2 w^3-5 v w^4-7 w^5)+u^3 (-7 v^5+28 v^4 w+19 v^3 w^2-8 v^2 w^3+5 v w^4-3 w^5))+b^4 c^6 v (2 v^2 (3 v-2 w) w^3 (v+w)^2+u^4 w (-5 v^3+8 v^2 w+10 v w^2-9 w^3)+u^5 (3 v^3-5 v^2 w-3 v w^2+3 w^3)+u v w^2 (5 v^4-6 v^3 w+12 v^2 w^2+22 v w^3-w^4)+u^3 (-3 v^5+5 v^4 w-8 v^3 w^2+19 v^2 w^3+28 v w^4-7 w^5)+u^2 w (-7 v^5-5 v^4 w+29 v^3 w^2+26 v^2 w^3+4 v w^4+5 w^5)))-a^6 u (b^8 (u+v)^3 w^3 (6 v w (v+w)+u (10 v^2+6 v w-w^2))+c^8 v^3 (u+w)^3 (6 v w (v+w)+u (-v^2+6 v w+10 w^2))-b^2 c^6 v^2 (u+w) (u^4 (-v^2+3 v w+12 w^2)-4 v w^2 (v^3-3 v^2 w-3 v w^2+w^3)+2 u v w (v^3+3 v^2 w+2 v w^2+6 w^3)+2 u^3 (v^3+v^2 w+7 v w^2+12 w^3)+u^2 (3 v^4-7 v^3 w+19 v^2 w^2+27 v w^3-8 w^4))-b^6 c^2 (u+v) w^2 (u^4 (12 v^2+3 v w-w^2)-4 v^2 w (v^3-3 v^2 w-3 v w^2+w^3)+2 u^3 (12 v^3+7 v^2 w+v w^2+w^3)+2 u v w (6 v^3+2 v^2 w+3 v w^2+w^3)+u^2 (-8 v^4+27 v^3 w+19 v^2 w^2-7 v w^3+3 w^4))+b^4 c^4 v w (u^5 (v^2-14 v w+w^2)+4 v^2 w^2 (-2 v^3+3 v^2 w+3 v w^2-2 w^3)-u^4 (9 v^3+10 v^2 w+10 v w^2+9 w^3)-u^3 (v^4-4 v^3 w+16 v^2 w^2-4 v w^3+w^4)+u v w (2 v^4+15 v^3 w+8 v^2 w^2+15 v w^3+2 w^4)+u^2 (9 v^5-8 v^4 w+24 v^3 w^2+24 v^2 w^3-8 v w^4+9 w^5))) : : (barys) 

Z2(P) = Z(NaNbNc, A"B"C") = a^4 v^2 w^2 (2 u+v+w)+a^2 u (b^2 w (u^2 (v-w)+v^2 (v+w)+u (2 v^2+2 v w-w^2))+c^2 v (u^2 (-v+w)+w^2 (v+w)+u (-v^2+2 v w+2 w^2)))-u (b^4 w (u^2 v+v (v+w)^2+u (2 v^2+3 v w+w^2))+c^4 v (u^2 w+w (v+w)^2+u (v^2+3 v w+2 w^2))-b^2 c^2 (2 v w (v+w)^2+u^2 (v^2+4 v w+w^2)+u (v^3+5 v^2 w+5 v w^2+w^3))) : : (barys)


*** Some points: 

P = I =X(1) 

Z1(P) = X(11570)

Z2(P) = X(5)

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P = G = X(2)

Z1(P) = (name pending)

 

= 12 a^12 b^2-46 a^10 b^4+64 a^8 b^6-54 a^6 b^8+32 a^4 b^10-8 a^2 b^12+12 a^12 c^2-40 a^10 b^2 c^2+8 a^8 b^4 c^2+65 a^6 b^6 c^2-88 a^4 b^8 c^2+47 a^2 b^10 c^2-4 b^12 c^2-46 a^10 c^4+8 a^8 b^2 c^4-10 a^6 b^4 c^4+32 a^4 b^6 c^4-82 a^2 b^8 c^4+14 b^10 c^4+64 a^8 c^6+65 a^6 b^2 c^6+32 a^4 b^4 c^6+86 a^2 b^6 c^6-10 b^8 c^6-54 a^6 c^8-88 a^4 b^2 c^8-82 a^2 b^4 c^8-10 b^6 c^8+32 a^4 c^10+47 a^2 b^2 c^10+14 b^4 c^10-8 a^2 c^12-4 b^2 c^12 : : (barys)

= (54 R^2 SB SC SW^2-18 SB SC SW^3+SB SW^4+SC SW^4)S^2 + (-54 R^2 SB SW-54 R^2 SC SW-18 R^2 SW^2+12 SB SW^2+12 SC SW^2-2 SW^3)S^4 + (108 R^2 + 27 SB + 27 SC - 18 SW)S^6 -2 SB SC SW^5 : : (barys) 

= lies on this line: {597,6055}

= (6-9-13)  search numbers:  [-1.59432999875996097621, -2.78522426654967345032, 6.30474128125413501929]

Z2(P) = X(20112)

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P = O = X(3)

Z1(P) = COMPLEMENT OF X(13289)  

= a^14 b^2-3 a^12 b^4+a^10 b^6+5 a^8 b^8-5 a^6 b^10-a^4 b^12+3 a^2 b^14-b^16+a^14 c^2-4 a^12 b^2 c^2+6 a^10 b^4 c^2-5 a^8 b^6 c^2+9 a^4 b^10 c^2-11 a^2 b^12 c^2+4 b^14 c^2-3 a^12 c^4+6 a^10 b^2 c^4-6 a^8 b^4 c^4+5 a^6 b^6 c^4-13 a^4 b^8 c^4+15 a^2 b^10 c^4-4 b^12 c^4+a^10 c^6-5 a^8 b^2 c^6+5 a^6 b^4 c^6+10 a^4 b^6 c^6-7 a^2 b^8 c^6-4 b^10 c^6+5 a^8 c^8-13 a^4 b^4 c^8-7 a^2 b^6 c^8+10 b^8 c^8-5 a^6 c^10+9 a^4 b^2 c^10+15 a^2 b^4 c^10-4 b^6 c^10-a^4 c^12-11 a^2 b^2 c^12-4 b^4 c^12+3 a^2 c^14+4 b^2 c^14-c^16 : : (barys)

= (84 R^4 + R^2 SB + R^2 SC - 37 R^2 SW + 4 SW^2)S^2 + 36 R^4 SB SC-27 R^2 SB SC SW+4 SB SC SW^2 : : (barys) 

= 3*X[2]-X[13289], X[3]+X[19506], X[4]+X[13293], X[66]+X[19140], X[110]+X[18381], 3*X[381]+X[2935], X[399]+3*X[1853], 5*X[1656]-X[10117], X[2892]+3*X[14561], 7*X[3090]+X[13203], X[3357]+X[7728], 9*X[5055]-X[9919], 2*X[5972]-X[10282], X[6759]-3*X[14643], X[9976]-3*X[23327], X[10264]-3*X[23332], X[10606]+X[23043], X[10733]-3*X[18376], X[12893]+X[18569], X[12901]-3*X[18281], X[14864]+2*X[16534], 5*X[15040]-X[17845], 5*X[20125]+3*X[32064]

= lies on these lines: {2,13289}, {3,19506}, {4,13293}, {5,1539}, {30,15090}, {66,19140}, {74,7577}, {110,18381}, {113,2072}, {125,389}, {265,13352}, {381,2935}, {399,1853}, {403,13202}, {427,7687}, {511,15116}, {542,23300}, {858,16163}, {974,18388}, {1503,10272}, {1511,14156}, {1568,12825}, {1656,10117}, {1899,12227}, {2778,9955}, {2781,6697}, {2892,14561}, {3043,25739}, {3090,13203}, {3357,7728}, {5055,9919}, {5092,12900}, {5094,19457}, {5097,20300}, {5448,5663}, {5576,23515}, {5972,10282}, {6759,14643}, {7579,17835}, {7722,23294}, {7741,10118}, {7951,19505}, {8889,18933}, {9976,23327}, {10024,16111}, {10113,12897}, {10254,11204}, {10264,23332}, {10576,13287}, {10577,13288}, {10606,23043}, {10721,16868}, {10733,18376}, {11572,17701}, {11597,15139}, {12219,23293}, {12358,21243}, {12893,18569}, {12901,18281}, {13371,17702}, {14864,16534}, {15040,17845}, {15051,31101}, {15472,18390}, {20125,32064}

= complement of X(13289)

= midpoint of X(i) and X(j) for these {i,j}: {3,19506}, {4,13293}, {5,23315}, {66,19140}, {110,18381}, {3357,7728},{12893,18569}, {15131,23325}

= reflection of X(i) in X(j) for these {i,j}: {10282,5972}, {12041,25563}, {20301,20300}

= (6-9-13)  search numbers:  [3.82671207454842953472, 3.11430682331060186765, -0.28156889171454351358]

Z2(P) = X(13561)

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P = X(5)

Z1(P) = X(5)X(49) ∩ X(1263)X(15038)

= (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (2 a^16-11 a^14 b^2+25 a^12 b^4-31 a^10 b^6+25 a^8 b^8-17 a^6 b^10+11 a^4 b^12-5 a^2 b^14+b^16-11 a^14 c^2+34 a^12 b^2 c^2-31 a^10 b^4 c^2-4 a^8 b^6 c^2+31 a^6 b^8 c^2-38 a^4 b^10 c^2+27 a^2 b^12 c^2-8 b^14 c^2+25 a^12 c^4-31 a^10 b^2 c^4-5 a^6 b^6 c^4+34 a^4 b^8 c^4-51 a^2 b^10 c^4+28 b^12 c^4-31 a^10 c^6-4 a^8 b^2 c^6-5 a^6 b^4 c^6-14 a^4 b^6 c^6+29 a^2 b^8 c^6-56 b^10 c^6+25 a^8 c^8+31 a^6 b^2 c^8+34 a^4 b^4 c^8+29 a^2 b^6 c^8+70 b^8 c^8-17 a^6 c^10-38 a^4 b^2 c^10-51 a^2 b^4 c^10-56 b^6 c^10+11 a^4 c^12+27 a^2 b^2 c^12+28 b^4 c^12-5 a^2 c^14-8 b^2 c^14+c^16) : : (barys)

= (4 R^2+8 SB+8 SC)S^4 + (-93 R^6+14 R^4 SB+14 R^4 SC+52 R^2 SB SC+76 R^4 SW-8 R^2 SB SW-8 R^2 SC SW-16 SB SC SW-16 R^2 SW^2)S^2 9 R^6 SB SC : : (barys) 

= lies on these lines: {5,49}, {1263,15038}, {6343,13621}, {6592,11584}, {10096,14071}, {14627,31674}, {14857,24306}

= barycentric quotient of X(i) and X(j) for these {i,j}: {24385,13582}

= trilinear product of X(i) and X(j) for these {i,j}: {1749,24385},{1749,24385}

= (6-9-13)  search numbers:  [0.24409911532262817460, 0.18909778573274509247, 3.39708950009741949698]

Z2(P) = X(5501)

 

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P = X(6)

Z1(P) = (name pending)

= 4 a^16-5 a^14 b^2-23 a^12 b^4+45 a^10 b^6-3 a^8 b^8-39 a^6 b^10+23 a^4 b^12-a^2 b^14-b^16-5 a^14 c^2+12 a^12 b^2 c^2-30 a^10 b^4 c^2-21 a^8 b^6 c^2+84 a^6 b^8 c^2-33 a^4 b^10 c^2-13 a^2 b^12 c^2+6 b^14 c^2-23 a^12 c^4-30 a^10 b^2 c^4+94 a^8 b^4 c^4-55 a^6 b^6 c^4-51 a^4 b^8 c^4+49 a^2 b^10 c^4-8 b^12 c^4+45 a^10 c^6-21 a^8 b^2 c^6-55 a^6 b^4 c^6+106 a^4 b^6 c^6-35 a^2 b^8 c^6-6 b^10 c^6-3 a^8 c^8+84 a^6 b^2 c^8-51 a^4 b^4 c^8-35 a^2 b^6 c^8+18 b^8 c^8-39 a^6 c^10-33 a^4 b^2 c^10+49 a^2 b^4 c^10-6 b^6 c^10+23 a^4 c^12-13 a^2 b^2 c^12-8 b^4 c^12-a^2 c^14+6 b^2 c^14-c^16 : : (barys)

= (162 R^4+27 R^2 SB+27 R^2 SC-99 R^2 SW-9 SB SW-9 SC SW+15 SW^2)S^4 + (-486 R^4 SB SC+243 R^2 SB SC SW+18 R^2 SB SW^2+18 R^2 SC SW^2-27 SB SC SW^2+6 R^2 SW^3-5 SB SW^3-5 SC SW^3-SW^4)S^2 + 18 R^2 SB SC SW^3-3 SB SC SW^4 : : (barys) 

= lies on this line: {541,597}

= (6-9-13)  search numbers:  [-0.31555309847327329277, -0.69853275929514409546, 4.26990397609945115048]

Z2(P) = X(20113)

Best regards,
Ercole Suppa