1a. Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P.
Let Oa be the center of the first circle in the Pappus chain in the arbelos determined by (B, A', C) [incircle of the arbelos] on the negative side of BC. Similarly Ob, Oc.
Which is the locus of P such that ABC, OaObOc are perspective?
[Antreas P. Hatzipolakis]:
1b. Let O1 be the center of the same circle for the arbelos (B, A', C) on the positive side of BC
ie O1 = the reflection of Oa in BC.
Similarly, O2, O3.
Which is the locus of P such that ABC, O1O2O3 are perspective?
2. Let O'a be the center of the Bankoff Archimedean circle (*), on the arbelos (B, A', C) on the negative side of BC and O'1 on the positive side (O'1 = the reflection of O'a in BC)
Which are the loci of P such that:
2a. ABC, O'aO'bO'c are pespective?
[Ercole Suppa]:
Let P(x:y:z) (barys)
*** 1a. The locus of points P such that ABC, OaObOc are perspective: {circumcubic through X(2), X(1131), X(1132)}
∑ (5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4+8 b^2 S-8 c^2 S) y^2 z-(5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4-8 b^2 S+8 c^2 S) y z^2 = 0 (barys)
*** Let W1 = W1(P). Some points:
W1(X(2)) = X(3590) = 1st DIXIT POINT
*** 2a. The locus of points P such that ABC, O1O2O3 are perspective: {circumcubic through X(2), X(1131), X(1132)}
∑ (5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4-8 b^2 S+8 c^2 S) y^2 z-(5 a^4-2 a^2 b^2-3 b^4-2 a^2 c^2+6 b^2 c^2-3 c^4+8 b^2 S-8 c^2 S) y z^2 = 0 (barys)
*** Let W2 = W2(P). Some points:
W2(X(2)) = X(3591) = 2nd DIXIT POINT
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*** 2a. The locus of points P such that ABC, O'aO'bO'c are perspective: {circumcubic through X(2), X(4), X(253), X(3316)}
∑ (a^2+b^2-c^2) (a^2-b^2+c^2+4 S) y^2 z-(a^2-b^2+c^2) (a^2+b^2-c^2+4 S) y z^2 = 0 (barys)
*** Let Q1 = Q1(P) the perspector. Some points:
Q1(X(2)) = ISOGONAL CONJUGATE OF X(6417)
= (a^2+b^2-c^2+8 S) (a^2-b^2+c^2+8 S) : : (barys)
= (4 S+SB) (4 S+SC) : : (barys)
= lies on these lines: {2,6418}, {4,6409}, {76,32813}, {2996,7376}, {3068,10194}, {5395,7375}, {7388,18845}
= isogonal conjugate of X(6417)
= isotomic conjugate of X(32812)
= barycentric quotient of X(i) and X(j) for these {i,j}: {2,32812}, {6,6417}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,6417}, {75,32812}
= (6-9-13) search numbers: [1.9752671063703917034, 1.5526369448749805525, 1.6541002402076733190]
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Q1(X(4)) = X(4)
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Q1(X(253)) = (name pending)
= (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4+8 a^2 S-8 b^2 S-8 c^2 S) : : (barys)
= (16 R^2-4 SW)S^3 + (-256 R^4 SB-256 R^4 SC-32 R^2 SB SC+256 R^4 SW+128 R^2 SB SW+128 R^2 SC SW+8 SB SC SW-128 R^2 SW^2-16 SB SW^2-16 SC SW^2+16 SW^3)S -128 R^4 SB SC-S^2 SB SC+48 R^2 SB SC SW-4 SB SC SW^2 : : (barys)
= lies on this line: {4,253}
= (6-9-13) search numbers: [12.0692312045613165807, 7.3866879560891270188, -7.0436108128748549472]
*** 2b. The locus of points P such that ABC, O'1O'2O'3 are perspective: {circumcubic through X(2), X(4), X(253), X(3317)}
∑ (a^2+b^2-c^2) (a^2-b^2+c^2-4 S) y^2 z-(a^2-b^2+c^2) (a^2+b^2-c^2-4 S) y z^2 = 0 (barys)
*** Let Q2 = Q2(P) the perspector. Some points:
Q2(X(2)) = ISOGONAL CONJUGATE OF X(6418)
= (a^2+b^2-c^2-8 S) (a^2-b^2+c^2-8 S) : : (barys)
= (4 S-SB) (4 S-SC) : : (barys)
= lies on these lines: {2,6417}, {4,6410}, {76,32812}, {2996,7375}, {3069,10195}, {5395,7376}, {5491,32807}, {7389,18845}
= isogonal conjugate of X(6418)
= isotomic conjugate of X(32813)
= barycentric quotient of X(i) and X(j) for these {i,j}: {2,32813},{6,6418}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,6418},{75,32813}
= (6-9-13) search numbers: [3.8879811209857682538, 1.6758357971730056244, 0.6860176434096360152]
----------------------
Q2(X(4)) = X(4)
----------------------
Q2(X(253)) = (name pending)
= (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4-8 a^2 S+8 b^2 S+8 c^2 S) : : (barys)
= (16 R^2-4 SW)S^3 + (-256 R^4 SB-256 R^4 SC-32 R^2 SB SC+256 R^4 SW+128 R^2 SB SW+128 R^2 SC SW+8 SB SC SW-128 R^2 SW^2-16 SB SW^2-16 SC SW^2+16 SW^3)S + 128 R^4 SB SC+S^2 SB SC-48 R^2 SB SC SW+4 SB SC SW^2 : : (barys)
= lies on this line: {4,253}
= (6-9-13) search numbers: [3.6469405728717216190, 0.7811890738902101701, 1.4166379355811313151]
Best regards
Ercole Suppa
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