[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of N.
Denote:
La, Lb, Lc = the parallels to AN, BN, CN through P.
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1, L2, L3
A'B'C', A*B*C* are parallelogic.
The parallelogic center (A'B'C', A*B*C*) is X(140)
The other one (A*B*C*, A'B'C') in terms of P?
Locus:
Which is the locus of P such that the parallelogic center (A*B*C*, A'B'C') lies on the Euler line?
Hi Antreas,
W = X(32551).
The line NW contains X{5,51,52,143,343,973,1154,1209,1568,3574,5562,5647,5891,9827,10095,10184,11583,11591,13364,13365,13565,14128,14140,14141,14531,14635,14845,15124,15537,15739,15961,15962,16336,16337,17167,18180,18874,18875,20424,21230,21265,21357,27355,27362,27374,30481,30482,31802,32352,32409,32551,33529,33530}.
Let a point, P, on NW = X[5] + t X[51]
then:
The parallelogic center(A*B*C*, A'B'C') = 3 a^2 (-a^2 + b^2 + c^2) + 4 S^2 t : :
Examples:
X(5) --> X(3)
X(51) --> X(2)
X(52) --> X(4)
X(143) --> X(5)
X(343) --> X(22)
X(973) --> X(13160)
X(1154) --> X(30)
X(1209) --> X(7512)
X(1568) --> X(2071)
X(3574) --> X(14118)
X(5562) --> X(20)
X(5891) --> X(376)
X(10095) --> X(140)
X(11591) --> X(550)
X(13364) --> X(549)
X(14128) --> X(548)
X(14531) --> X(3146)
X(14845) --> X(3524)
X(17167) --> X(4184)
X(18180) --> X(21)
X(18874) --> X(3530)
X(20424) --> X(14130)
X(21230) --> X(13564)
X(27355) --> X(15717)
X(27374) --> X(384)
X(31802) --> X(1593)
------------------------------
X(9827) ->
------------------------------
X(10184) -->
------------------------------
X(13365) -->
------------------------------
X(15739) -->
------------------------------
X(21357) -->
------------------------------
X(32352) -->
------------------------------
X(33529) -->
------------------------------
X(33530) -->
------------------------------
Best regards,
Peter Moses
Denote:
La, Lb, Lc = the parallels to AN, BN, CN through P.
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1, L2, L3
A'B'C', A*B*C* are parallelogic.
The parallelogic center (A'B'C', A*B*C*) is X(140)
The other one (A*B*C*, A'B'C') in terms of P?
Locus:
Which is the locus of P such that the parallelogic center (A*B*C*, A'B'C') lies on the Euler line?
I think it is the line NW, where W is the point of concurrence of the parallels to AN, BN, CN through Na, Nb, Nc, resp.
where Na, Nb, Nc are the NPC centers of NBC, NCA, NAB, resp.
That is:
Let ABC be a triangle and A'B'C' the pedal triangle of N.
Denote:
Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.
W = the point of concurrence of the parallels to AN, BN, CN through Na, Nb, Nc, resp.
P = a point on the line NW.
La, Lb, Lc = the parallels though P to AN, BN, CN, resp,
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1, L2, L3
A'B'C', A*B*C* are parallelogic.
The parallelogic center (A'B'C', A*B*C*) is X(140)
The parallelogic center of (A*B*C*, A'B'C') lies on the Euler line of ABC.
Question:
Which triangle centers lie on the line NW and which are the respective parallelogic centers (A*B*C*, A'B'C') on the Euler line?
[Peter Moses]:
where Na, Nb, Nc are the NPC centers of NBC, NCA, NAB, resp.
That is:
Let ABC be a triangle and A'B'C' the pedal triangle of N.
Denote:
Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.
W = the point of concurrence of the parallels to AN, BN, CN through Na, Nb, Nc, resp.
P = a point on the line NW.
La, Lb, Lc = the parallels though P to AN, BN, CN, resp,
L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1, L2, L3
A'B'C', A*B*C* are parallelogic.
The parallelogic center (A'B'C', A*B*C*) is X(140)
The parallelogic center of (A*B*C*, A'B'C') lies on the Euler line of ABC.
Question:
Which triangle centers lie on the line NW and which are the respective parallelogic centers (A*B*C*, A'B'C') on the Euler line?
[Peter Moses]:
Hi Antreas,
W = X(32551).
The line NW contains X{5,51,52,143,343,973,1154,1209,1568,3574,5562,5647,5891,9827,10095,10184,11583,11591,13364,13365,13565,14128,14140,14141,14531,14635,14845,15124,15537,15739,15961,15962,16336,16337,17167,18180,18874,18875,20424,21230,21265,21357,27355,27362,27374,30481,30482,31802,32352,32409,32551,33529,33530}.
Let a point, P, on NW = X[5] + t X[51]
then:
The parallelogic center(A*B*C*, A'B'C') = 3 a^2 (-a^2 + b^2 + c^2) + 4 S^2 t : :
Examples:
X(5) --> X(3)
X(51) --> X(2)
X(52) --> X(4)
X(143) --> X(5)
X(343) --> X(22)
X(973) --> X(13160)
X(1154) --> X(30)
X(1209) --> X(7512)
X(1568) --> X(2071)
X(3574) --> X(14118)
X(5562) --> X(20)
X(5891) --> X(376)
X(10095) --> X(140)
X(11591) --> X(550)
X(13364) --> X(549)
X(14128) --> X(548)
X(14531) --> X(3146)
X(14845) --> X(3524)
X(17167) --> X(4184)
X(18180) --> X(21)
X(18874) --> X(3530)
X(20424) --> X(14130)
X(21230) --> X(13564)
X(27355) --> X(15717)
X(27374) --> X(384)
X(31802) --> X(1593)
------------------------------
X(9827) ->
(a^2 - b^2 - c^2)*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 3*a^2*b^6 - b^8 - 3*a^6*c^2 - 10*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 4*b^6*c^2 - a^4*c^4 - 3*a^2*b^2*c^4 - 6*b^4*c^4 + 3*a^2*c^6 + 4*b^2*c^6 - c^8) : :
= lies on these lines: {2,3}, {17,11516}, {18,11515}, {68,3796}, {141,10539}, {216,7755}, {323,2889}, {397,10635}, {398,10634}, {511,12242}, {524,13431}, {567,13142}, {1147,13394}, {1176,3519}, {1209,1503}, {1506,22052}, {1614,31831}, {3917,9820}, {5012,13292}, {5447,11064}, {5462,32269}, {5596,14530}, {5882,24301}, {5891,16252}, {5907,14862}, {6000,32348}, {6696,14855}, {7746,10979}, {8550,19131}, {8960,11514}, {10625,23292}, {10984,12359}, {11574,25555}, {11649,13433}, {11695,32223}, {11803,12606}, {11898,19119}, {12241,18555}, {12358,16534}, {13336,13567}, {14864,21243}, {29317,32396}
------------------------------
X(10184) -->
a^2*(a^2 - b^2 - c^2)*(3*a^6*b^2 - 6*a^4*b^4 + 3*a^2*b^6 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 2*b^6*c^2 - 6*a^4*c^4 - 3*a^2*b^2*c^4 - 4*b^4*c^4 + 3*a^2*c^6 + 2*b^2*c^6) : :
= lies on these lines: {2,3},{216,14773},{511,32078},{3289,22052},{3928,26900},{3929,26901},{13409,26907},{15107,31626}
------------------------------
X(13365) -->
2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 - 2*a^6*b^2*c^2 + 7*a^4*b^4*c^2 + 3*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 7*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 3*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :
= lies on thse lines: {2,3}, {511,8254}, {1216,15806}, {1503,14076}, {3796,18356}, {5012,32165}, {5663,32348}, {6689,13391}, {10272,11793}, {11592,14156}, {32205,32223}
------------------------------
X(15739) -->
4*a^10 - 7*a^8*b^2 - 2*a^6*b^4 + 8*a^4*b^6 - 2*a^2*b^8 - b^10 - 7*a^8*c^2 + 14*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 4*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 2*b^6*c^4 + 8*a^4*c^6 - 6*a^2*b^2*c^6 - 2*b^4*c^6 - 2*a^2*c^8 + 3*b^2*c^8 - c^10 : :
= lies on these lines: {2,3}, {1503,15062}, {3580,13403}, {3629,5889}, {3796,5925}, {5878,6800}, {5894,6293}, {6146,11440}, {6329,13434}, {7689,12022}, {9628,15326}, {11557,16111}, {11793,16163}, {12383,31831}, {12825,24981}, {14810,26156}
------------------------------
X(21357) -->
a^2*(3*a^8 - 6*a^6*b^2 + 6*a^2*b^6 - 3*b^8 - 6*a^6*c^2 - 5*a^4*b^2*c^2 + 7*a^2*b^4*c^2 + 4*b^6*c^2 + 7*a^2*b^2*c^4 - 2*b^4*c^4 + 6*a^2*c^6 + 4*b^2*c^6 - 3*c^8) : :
= lies on these lines: {2,3}, {542,2916}, {1154,6030}, {1350,9703}, {10610,13482}, {13353,21849}, {14627,21969}, {15037,22352}, {15038,15107}, {15080,15087}
------------------------------
X(32352) -->
a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 5*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 3*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
= lies on these lines: {2,3}, {49,12383}, {54,17702}, {146,12162}, {184,12278}, {185,3448}, {265,13630}, {316,26166}, {511,32338}, {542,9972}, {1199,12370}, {1204,23293}, {1235,13219}, {1531,11793}, {1994,12233}, {2777,14076}, {2888,13754}, {2892,29959}, {3410,12111}, {3519,18363}, {3521,5663}, {3567,7706}, {3580,13568}, {3818,11439}, {4846,11457}, {5012,21659}, {5562,15108}, {5890,9927}, {5894,13203}, {6000,32337}, {6241,18474}, {6800,17845}, {7592,12293}, {7730,15103}, {7748,26216}, {9539,11392}, {9545,12118}, {9705,30714}, {9729,13851}, {9827,22948}, {10113,12006}, {10574,18392}, {10733,13403}, {11003,19467}, {11064,22555}, {11440,21243}, {11550,12279}, {11746,22466}, {12254,30522}, {13391,15800}, {13579,31363}, {13585,13599}, {14516,14683}, {15035,19479}, {15043,18390}, {15055,19506}, {15072,18381}, {15305,22802}, {15740,18434}, {18376,20791}, {18380,20792}, {18382,25406}, {18387,18917}, {18553,32247}, {29012,32340}, {32273,33749}, {32353,32369}
------------------------------
X(33529) -->
a^2*(Sqrt[3]*(a^4 - b^4 - c^4) + 2*(a^2 - b^2 - c^2)*S) : :
= lies on these lines: {2,3}, {13,16463}, {15,1337}, {16,15080}, {61,3060}, {62,5012}, {110,14538}, {184,11126}, {216,11421}, {302,33801}, {396,11142}, {511,11127}, {524,14173}, {577,11420}, {616,2925}, {617,1606}, {621,2923}, {627,1607}, {1495,11131}, {1993,5864}, {2004,5238}, {2979,14540}, {3098,11130}, {3442,22113}, {3455,22570}, {3796,22238}, {5237,6030}, {5615,11003}, {5873,19779}, {6582,14181}, {9736,14169}, {10409,16642}, {11542,21310}, {14705,30560}, {22236,33586}
------------------------------
X(33530) -->
a^2*(Sqrt[3]*(a^4 - b^4 - c^4) - 2*(a^2 - b^2 - c^2)*S) : :
= lies on these lines: {2,3}, {14,16464}, {15,15080}, {16,1338}, {61,5012}, {62,3060}, {110,14539}, {184,11127}, {216,11420}, {303,33801}, {395,11141}, {511,11126}, {524,14179}, {577,11421}, {616,1605}, {617,2926}, {622,2924}, {628,1608}, {1495,11130}, {1993,5865}, {2005,5237}, {2979,14541}, {3098,11131}, {3443,22114}, {3455,22568}, {3796,22236}, {5238,6030}, {5611,11003}, {5872,19778}, {6295,14177}, {9735,14170}, {10410,16643}, {11543,21311}, {14704,30559}, {22238,33586}
------------------------------
Best regards,
Peter Moses
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