Let ABC be a triangle, (D), (E), (F) the A-, B-, C- mixtilinear incircles, resp. and M a point on the circumcircle.
Denote:
X, Y, Z = the touchpoints of the circumcircle and (D), (E), (F), resp.
X' = the other than X intersection of (D) and MX
Y' = the other than Y intersection of (E) and MY
Z' = the other than Z intersection of (F) and MZ
Then ABC, X'Y'Z' are perspective.
As M moves on the circumcircle the perspector moves on the incircle.
[APH]
In general, let M be a variable point.
Which is the locus of M such that ABC, X'Y'Z' are perspective ?
In general, let M be a variable point.
Which is the locus of M such that ABC, X'Y'Z' are perspective ?
[Ercole Suppa]:
Let M(x:y:z) (barys) and Q = Q(M) the perspector of ABC and X'Y'Z'
*** The locus of points M such that ABC, X'Y'Z' are perspective is
{q1 = trilinear polar of X(651)} ∪ {q2 = circumcircle} ∪ {q3 = cubic passing through X(5217)}
q1: ∑ b (a - b - c) (b - c) c x = 0
q2: ∑ a^2 y z = 0
q3: 4 a^2 b^4 c^3 x^3-8 a b^5 c^3 x^3+4 b^6 c^3 x^3+4 a^2 b^3 c^4 x^3-16 a b^4 c^4 x^3+12 b^5 c^4 x^3-8 a b^3 c^5 x^3+12 b^4 c^5 x^3+4 b^3 c^6 x^3+5 a^4 b^2 c^3 x^2 y-16 a^3 b^3 c^3 x^2 y+18 a^2 b^4 c^3 x^2 y-8 a b^5 c^3 x^2 y+b^6 c^3 x^2 y+3 a^4 b c^4 x^2 y-6 a^3 b^2 c^4 x^2 y+12 a^2 b^3 c^4 x^2 y-10 a b^4 c^4 x^2 y+b^5 c^4 x^2 y+2 a^3 b c^5 x^2 y-10 a^2 b^2 c^5 x^2 y+2 a b^3 c^5 x^2 y-2 b^4 c^5 x^2 y-4 a^2 b c^6 x^2 y+2 a b^2 c^6 x^2 y-2 b^3 c^6 x^2 y-2 a b c^7 x^2 y+b^2 c^7 x^2 y+b c^8 x^2 y+a^6 c^3 x y^2-8 a^5 b c^3 x y^2+18 a^4 b^2 c^3 x y^2-16 a^3 b^3 c^3 x y^2+5 a^2 b^4 c^3 x y^2+a^5 c^4 x y^2-10 a^4 b c^4 x y^2+12 a^3 b^2 c^4 x y^2-6 a^2 b^3 c^4 x y^2+3 a b^4 c^4 x y^2-2 a^4 c^5 x y^2+2 a^3 b c^5 x y^2-10 a^2 b^2 c^5 x y^2+2 a b^3 c^5 x y^2-2 a^3 c^6 x y^2+2 a^2 b c^6 x y^2-4 a b^2 c^6 x y^2+a^2 c^7 x y^2-2 a b c^7 x y^2+a c^8 x y^2+4 a^6 c^3 y^3-8 a^5 b c^3 y^3+4 a^4 b^2 c^3 y^3+12 a^5 c^4 y^3-16 a^4 b c^4 y^3+4 a^3 b^2 c^4 y^3+12 a^4 c^5 y^3-8 a^3 b c^5 y^3+4 a^3 c^6 y^3+3 a^4 b^4 c x^2 z+2 a^3 b^5 c x^2 z-4 a^2 b^6 c x^2 z-2 a b^7 c x^2 z+b^8 c x^2 z+5 a^4 b^3 c^2 x^2 z-6 a^3 b^4 c^2 x^2 z-10 a^2 b^5 c^2 x^2 z+2 a b^6 c^2 x^2 z+b^7 c^2 x^2 z-16 a^3 b^3 c^3 x^2 z+12 a^2 b^4 c^3 x^2 z+2 a b^5 c^3 x^2 z-2 b^6 c^3 x^2 z+18 a^2 b^3 c^4 x^2 z-10 a b^4 c^4 x^2 z-2 b^5 c^4 x^2 z-8 a b^3 c^5 x^2 z+b^4 c^5 x^2 z+b^3 c^6 x^2 z+4 a^6 b^2 c x y z-8 a^4 b^4 c x y z+4 a^2 b^6 c x y z+4 a^6 b c^2 x y z-2 a^5 b^2 c^2 x y z+6 a^4 b^3 c^2 x y z+6 a^3 b^4 c^2 x y z-2 a^2 b^5 c^2 x y z+4 a b^6 c^2 x y z+6 a^4 b^2 c^3 x y z-36 a^3 b^3 c^3 x y z+6 a^2 b^4 c^3 x y z-8 a^4 b c^4 x y z+6 a^3 b^2 c^4 x y z+6 a^2 b^3 c^4 x y z-8 a b^4 c^4 x y z-2 a^2 b^2 c^5 x y z+4 a^2 b c^6 x y z+4 a b^2 c^6 x y z+a^8 c y^2 z-2 a^7 b c y^2 z-4 a^6 b^2 c y^2 z+2 a^5 b^3 c y^2 z+3 a^4 b^4 c y^2 z+a^7 c^2 y^2 z+2 a^6 b c^2 y^2 z-10 a^5 b^2 c^2 y^2 z-6 a^4 b^3 c^2 y^2 z+5 a^3 b^4 c^2 y^2 z-2 a^6 c^3 y^2 z+2 a^5 b c^3 y^2 z+12 a^4 b^2 c^3 y^2 z-16 a^3 b^3 c^3 y^2 z-2 a^5 c^4 y^2 z-10 a^4 b c^4 y^2 z+18 a^3 b^2 c^4 y^2 z+a^4 c^5 y^2 z-8 a^3 b c^5 y^2 z+a^3 c^6 y^2 z+a^6 b^3 x z^2+a^5 b^4 x z^2-2 a^4 b^5 x z^2-2 a^3 b^6 x z^2+a^2 b^7 x z^2+a b^8 x z^2-8 a^5 b^3 c x z^2-10 a^4 b^4 c x z^2+2 a^3 b^5 c x z^2+2 a^2 b^6 c x z^2-2 a b^7 c x z^2+18 a^4 b^3 c^2 x z^2+12 a^3 b^4 c^2 x z^2-10 a^2 b^5 c^2 x z^2-4 a b^6 c^2 x z^2-16 a^3 b^3 c^3 x z^2-6 a^2 b^4 c^3 x z^2+2 a b^5 c^3 x z^2+5 a^2 b^3 c^4 x z^2+3 a b^4 c^4 x z^2+a^8 b y z^2+a^7 b^2 y z^2-2 a^6 b^3 y z^2-2 a^5 b^4 y z^2+a^4 b^5 y z^2+a^3 b^6 y z^2-2 a^7 b c y z^2+2 a^6 b^2 c y z^2+2 a^5 b^3 c y z^2-10 a^4 b^4 c y z^2-8 a^3 b^5 c y z^2-4 a^6 b c^2 y z^2-10 a^5 b^2 c^2 y z^2+12 a^4 b^3 c^2 y z^2+18 a^3 b^4 c^2 y z^2+2 a^5 b c^3 y z^2-6 a^4 b^2 c^3 y z^2-16 a^3 b^3 c^3 y z^2+3 a^4 b c^4 y z^2+5 a^3 b^2 c^4 y z^2+4 a^6 b^3 z^3+12 a^5 b^4 z^3+12 a^4 b^5 z^3+4 a^3 b^6 z^3-8 a^5 b^3 c z^3-16 a^4 b^4 c z^3-8 a^3 b^5 c z^3+4 a^4 b^3 c^2 z^3+4 a^3 b^4 c^2 z^3 = 0 (barys)
*** Pairs (M = X(i), Q(M) = X(j))
- Pairs (M = X(i) ∈ q1, Q(M) = X(j)): {1,8}, {36,1318}, {56,56}, {999,55}, {1381,2447}, {1382,2446}, {1420,1118}, {5563,60}, {13462,479}
- Pairs (M = X(i) ∈ q2, Q(M) = X(j)): {74,3024}, {98,3023}, {99,3027}, {100,1317}, {101,1362}, {102,1364}, {103,3022}, {104,11}, {105,1358}, {106,1357}, {107,3324}, {108,1359}, {109,1361}, {110,3028}, {111,3325}, {112,3320}, {689,7334}, {691,6023}, {729,7333}, {741,1356}, {759,1365}, {842,6027}, {901,13756}, {930,7159}, {934,1360}, {953,3025}, {1141,3327}, {1290,31524}, {1292,3021}, {1293,6018}, {1294,7158}, {1295,3318}, {1296,6019}, {1297,6020}, {1308,3322}, {1381,2447}, {1382,2446}, {2222,3319}, {2687,31522}, {2716,3326}, {2717,3328}, {2718,14027}, {2725,3323},{3659,10506}, {7597,10501}, {8691,1366}, {10496,10505}, {10497,10491}, {12032,15615}, {14074,3321}, {26700,1354},{26701,1363}, {26702,1367}, {28233,5577}, {28291,5580}, {28293,6024}, {28474,6021}, {28848,31892}, {29055,1355}, {29310,3026}
*** Some points
Q(X(3)) = ISOGONAL CONJUGATE OF X(6049)
= a^2 (a+b-3 c)^2 (a-b-c) (a-3 b+c)^2 : : (barys)
= lies on these lines: {11,6556}, {55,3445}, {200,2098}, {220,8163}, {1043,4345}, {1293,5204}, {3057,8056}, {3304,14261}, {7962,10563}, {12513,27834}, {16078,21453}
= barycentric product of X(i) and X(j) for these {i,j}: {220,16078}, {346,16079}, {3445,6557}, {3680,8056}
= barycentric quotient of X(i) and X(j) for these {i,j}: {6,6049}, {220,15519}, {657,4943}, {663,31182}, {3445,5435},{3680,18743}, {16079,279}
= trilinear product of X(i) and X(j) for these {i,j}: {200,16079}, {1253,16078}, {1253,16078}, {3445,3680}, {6556,16945}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,6049}, {200,15519}, {3900,4943}
= (6-9-13) search numbers: [5.7667386067951693818, 1.3515793609676178239, 0.0433840288705655104]
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Q(X(476)) = X(1)X(16168) ∩ X(11)X(25641)
= (a+b-c) (a-b+c) (b+c)^2 (a^6-2 a^4 b^2+a^2 b^4+2 a^4 b c-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)^2 : : (barys)
= lies on these lines: {1,16168}, {11,25641}, {12,3258}, {30,3024}, {55,477}, {56,476}, {388,14731}, {496,18319}, {523,3028}, {1478,20957}, {3023,7286}, {3025,24470}, {3649,31522}, {5432,31379}, {5433,22104}, {5434,6027}, {7158,10149}, {12091,18447}, {12903,17511}, {12953,14989}
= (6-9-13) search numbers: [0.1384266026106665735, 0.3559422884886237418, 3.3303536963642550172]
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Q(X(477)) = X(1)X(16168) ∩ X(11)X(3258)
= (b+c-a) (b-c)^2 (a^6-2 a^4 b^2+a^2 b^4-2 a^4 b c+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)^2 : : (barys)
= lies on these lines: {1,16168}, {11,3258}, {12,25641}, {30,3028}, {55,476}, {56,477}, {495,18319}, {497,14731}, {523,3024}, {1479,20957}, {3027,5160}, {3058,6023}, {5432,22104}, {5433,31379}, {10543,31524}, {12091,18455}, {12904,17511}, {12943,14989}
= (6-9-13) search numbers: [3.2421904163033997365, 3.0246747304254425683, 0.0502633225498112928]
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Q(X(675)) = REFLECTION OF X(6025) IN X(1)
= (b-c)^2 (a-b+c) (a+b-c) (-a^3+a^2 b+a^2 c+b^2 c+b c^2)^2 : : (barys)
= 2*X[1]-X[6025]
= lies on the incircle and these lines: {1,6025}, {11,25642}, {12,5513}, {56,675}, {544,1362}, {5433,31380}
= reflection of X(6025) in X(1)
= (6-9-13) search numbers: [1.8817526473636784786, 3.2544340540906360689, 0.5190935302922409883]
Best regards,
Ercole Suppa
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