CYCLOLOGIC

Let ABC be a triangle

Denote

1. Oa, Ob, Oc = the circumcenters of HBC, HCA, HAB, resp.

ABC, OaObOc are cyclologic, since Oa, Ob, Oc are the reflections of O in BC,CA,AB, resp.
Cyclologic center (OaObOc, ABC) = antigonal conjugate of O = X(265)

2. Sa, Sb, Sc = the X(54) of HBC, HCA, HAB, resp.

ABC,SaSbSc are cyclologic

Cyclologic centers?

ETC

X(72513) = (name pending)

Barycentrics    a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-5 a^8 b^2 c^2+4 a^6 b^4 c^2+a^4 b^6 c^2+a^2 b^8 c^2-2 b^10 c^2-3 a^8 c^4+4 a^6 b^2 c^4+3 a^4 b^4 c^4+2 a^2 b^6 c^4-b^8 c^4+2 a^6 c^6+a^4 b^2 c^6+2 a^2 b^4 c^6+4 b^6 c^6+2 a^4 c^8+a^2 b^2 c^8-b^4 c^8-3 a^2 c^10-2 b^2 c^10+c^12) (a^20-2 a^18 b^2-3 a^16 b^4+8 a^14 b^6+2 a^12 b^8-12 a^10 b^10+2 a^8 b^12+8 a^6 b^14-3 a^4 b^16-2 a^2 b^18+b^20-5 a^18 c^2+5 a^16 b^2 c^2+9 a^14 b^4 c^2-7 a^12 b^6 c^2-2 a^10 b^8 c^2-2 a^8 b^10 c^2-7 a^6 b^12 c^2+9 a^4 b^14 c^2+5 a^2 b^16 c^2-5 b^18 c^2+8 a^16 c^4-5 a^14 b^2 c^4+2 a^12 b^4 c^4+2 a^10 b^6 c^4-14 a^8 b^8 c^4+2 a^6 b^10 c^4+2 a^4 b^12 c^4-5 a^2 b^14 c^4+8 b^16 c^4+16 a^12 b^2 c^6-8 a^10 b^4 c^6-8 a^8 b^6 c^6-8 a^6 b^8 c^6-8 a^4 b^10 c^6+16 a^2 b^12 c^6-14 a^12 c^8-39 a^10 b^2 c^8-28 a^8 b^4 c^8-27 a^6 b^6 c^8-28 a^4 b^8 c^8-39 a^2 b^10 c^8-14 b^12 c^8+14 a^10 c^10+32 a^8 b^2 c^10+35 a^6 b^4 c^10+35 a^4 b^6 c^10+32 a^2 b^8 c^10+14 b^10 c^10+5 a^6 b^2 c^12+8 a^4 b^4 c^12+5 a^2 b^6 c^12-8 a^6 c^14-20 a^4 b^2 c^14-20 a^2 b^4 c^14-8 b^6 c^14+5 a^4 c^16+9 a^2 b^2 c^16+5 b^4 c^16-a^2 c^18-b^2 c^18) (a^20-5 a^18 b^2+8 a^16 b^4-14 a^12 b^8+14 a^10 b^10-8 a^6 b^14+5 a^4 b^16-a^2 b^18-2 a^18 c^2+5 a^16 b^2 c^2-5 a^14 b^4 c^2+16 a^12 b^6 c^2-39 a^10 b^8 c^2+32 a^8 b^10 c^2+5 a^6 b^12 c^2-20 a^4 b^14 c^2+9 a^2 b^16 c^2-b^18 c^2-3 a^16 c^4+9 a^14 b^2 c^4+2 a^12 b^4 c^4-8 a^10 b^6 c^4-28 a^8 b^8 c^4+35 a^6 b^10 c^4+8 a^4 b^12 c^4-20 a^2 b^14 c^4+5 b^16 c^4+8 a^14 c^6-7 a^12 b^2 c^6+2 a^10 b^4 c^6-8 a^8 b^6 c^6-27 a^6 b^8 c^6+35 a^4 b^10 c^6+5 a^2 b^12 c^6-8 b^14 c^6+2 a^12 c^8-2 a^10 b^2 c^8-14 a^8 b^4 c^8-8 a^6 b^6 c^8-28 a^4 b^8 c^8+32 a^2 b^10 c^8-12 a^10 c^10-2 a^8 b^2 c^10+2 a^6 b^4 c^10-8 a^4 b^6 c^10-39 a^2 b^8 c^10+14 b^10 c^10+2 a^8 c^12-7 a^6 b^2 c^12+2 a^4 b^4 c^12+16 a^2 b^6 c^12-14 b^8 c^12+8 a^6 c^14+9 a^4 b^2 c^14-5 a^2 b^4 c^14-3 a^4 c^16+5 a^2 b^2 c^16+8 b^4 c^16-2 a^2 c^18-5 b^2 c^18+c^20 : :

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9658.

X(72513) lies on this line: {4, 72514}

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X(72514) = (name pending)

Barycentrics    a^46-13 a^44 b^2+70 a^42 b^4-190 a^40 b^6+206 a^38 b^8+252 a^36 b^10-1093 a^34 b^12+1119 a^32 b^14+646 a^30 b^16-2598 a^28 b^18+1808 a^26 b^20+1352 a^24 b^22-2912 a^22 b^24+1052 a^20 b^26+1378 a^18 b^28-1526 a^16 b^30+177 a^14 b^32+587 a^12 b^34-382 a^10 b^36+14 a^8 b^38+90 a^6 b^40-48 a^4 b^42+11 a^2 b^44-b^46-13 a^44 c^2+142 a^42 b^2 c^2-629 a^40 b^4 c^2+1353 a^38 b^6 c^2-1007 a^36 b^8 c^2-1584 a^34 b^10 c^2+4121 a^32 b^12 c^2-2528 a^30 b^14 c^2-1826 a^28 b^16 c^2+3224 a^26 b^18 c^2-1482 a^24 b^20 c^2+858 a^22 b^22 c^2-494 a^20 b^24 c^2-2132 a^18 b^26 c^2+3498 a^16 b^28 c^2-632 a^14 b^30 c^2-2497 a^12 b^32 c^2+2138 a^10 b^34 c^2-145 a^8 b^36 c^2-715 a^6 b^38 c^2+461 a^4 b^40 c^2-124 a^2 b^42 c^2+13 b^44 c^2+70 a^42 c^4-629 a^40 b^2 c^4+2245 a^38 b^4 c^4-3815 a^36 b^6 c^4+2361 a^34 b^8 c^4+1719 a^32 b^10 c^4-3563 a^30 b^12 c^4+2927 a^28 b^14 c^4-3682 a^26 b^16 c^4+2824 a^24 b^18 c^4+2669 a^22 b^20 c^4-5845 a^20 b^22 c^4+3126 a^18 b^24 c^4+54 a^16 b^26 c^4-2301 a^14 b^28 c^4+4745 a^12 b^30 c^4-4040 a^10 b^32 c^4-23 a^8 b^34 c^4+2518 a^6 b^36 c^4-1884 a^4 b^38 c^4+597 a^2 b^40 c^4-73 b^42 c^4-190 a^40 c^6+1353 a^38 b^2 c^6-3815 a^36 b^4 c^6+5616 a^34 b^6 c^6-5525 a^32 b^8 c^6+4978 a^30 b^10 c^6-1681 a^28 b^12 c^6-5697 a^26 b^14 c^6+8248 a^24 b^16 c^6-3793 a^22 b^18 c^6+2189 a^20 b^20 c^6-1518 a^18 b^22 c^6-4692 a^16 b^24 c^6+9020 a^14 b^26 c^6-6475 a^12 b^28 c^6+1627 a^10 b^30 c^6+2946 a^8 b^32 c^6-5446 a^6 b^34 c^6+4166 a^4 b^36 c^6-1532 a^2 b^38 c^6+221 b^40 c^6+206 a^38 c^8-1007 a^36 b^2 c^8+2361 a^34 b^4 c^8-5525 a^32 b^6 c^8+11421 a^30 b^8 c^8-12135 a^28 b^10 c^8+2684 a^26 b^12 c^8+3660 a^24 b^14 c^8-1518 a^22 b^16 c^8+1420 a^20 b^18 c^8-3180 a^18 b^20 c^8+5462 a^16 b^22 c^8-9713 a^14 b^24 c^8+6227 a^12 b^26 c^8+4814 a^10 b^28 c^8-10446 a^8 b^30 c^8+8756 a^6 b^32 c^8-5113 a^4 b^34 c^8+1961 a^2 b^36 c^8-335 b^38 c^8+252 a^36 c^10-1584 a^34 b^2 c^10+1719 a^32 b^4 c^10+4978 a^30 b^6 c^10-12135 a^28 b^8 c^10+8432 a^26 b^10 c^10-3073 a^24 b^12 c^10+2820 a^22 b^14 c^10-907 a^20 b^16 c^10+1013 a^18 b^18 c^10-3551 a^16 b^20 c^10+4123 a^14 b^22 c^10-1368 a^12 b^24 c^10-9053 a^10 b^26 c^10+16922 a^8 b^28 c^10-11293 a^6 b^30 c^10+2794 a^4 b^32 c^10-76 a^2 b^34 c^10-13 b^36 c^10-1093 a^34 c^12+4121 a^32 b^2 c^12-3563 a^30 b^4 c^12-1681 a^28 b^6 c^12+2684 a^26 b^8 c^12-3073 a^24 b^10 c^12+5021 a^22 b^12 c^12-1951 a^20 b^14 c^12-320 a^18 b^16 c^12+1007 a^16 b^18 c^12-674 a^14 b^20 c^12-3692 a^12 b^22 c^12+9574 a^10 b^24 c^12-14546 a^8 b^26 c^12+9916 a^6 b^28 c^12+966 a^4 b^30 c^12-3785 a^2 b^32 c^12+1089 b^34 c^12+1119 a^32 c^14-2528 a^30 b^2 c^14+2927 a^28 b^4 c^14-5697 a^26 b^6 c^14+3660 a^24 b^8 c^14+2820 a^22 b^10 c^14-1951 a^20 b^12 c^14+350 a^18 b^14 c^14-252 a^16 b^16 c^14-263 a^14 b^18 c^14+3278 a^12 b^20 c^14-8167 a^10 b^22 c^14+7018 a^8 b^24 c^14-2327 a^6 b^26 c^14-3550 a^4 b^28 c^14+5552 a^2 b^30 c^14-1989 b^32 c^14+646 a^30 c^16-1826 a^28 b^2 c^16-3682 a^26 b^4 c^16+8248 a^24 b^6 c^16-1518 a^22 b^8 c^16-907 a^20 b^10 c^16-320 a^18 b^12 c^16-252 a^16 b^14 c^16+526 a^14 b^16 c^16-805 a^12 b^18 c^16+4498 a^10 b^20 c^16-2940 a^8 b^22 c^16-5870 a^6 b^24 c^16+5734 a^4 b^26 c^16-2450 a^2 b^28 c^16+918 b^30 c^16-2598 a^28 c^18+3224 a^26 b^2 c^18+2824 a^24 b^4 c^18-3793 a^22 b^6 c^18+1420 a^20 b^8 c^18+1013 a^18 b^10 c^18+1007 a^16 b^12 c^18-263 a^14 b^14 c^18-805 a^12 b^16 c^18-2018 a^10 b^18 c^18+1200 a^8 b^20 c^18+7493 a^6 b^22 c^18-7028 a^4 b^24 c^18-2104 a^2 b^26 c^18+2210 b^28 c^18+1808 a^26 c^20-1482 a^24 b^2 c^20+2669 a^22 b^4 c^20+2189 a^20 b^6 c^20-3180 a^18 b^8 c^20-3551 a^16 b^10 c^20-674 a^14 b^12 c^20+3278 a^12 b^14 c^20+4498 a^10 b^16 c^20+1200 a^8 b^18 c^20-6244 a^6 b^20 c^20+3502 a^4 b^22 c^20+3666 a^2 b^24 c^20-4250 b^26 c^20+1352 a^24 c^22+858 a^22 b^2 c^22-5845 a^20 b^4 c^22-1518 a^18 b^6 c^22+5462 a^16 b^8 c^22+4123 a^14 b^10 c^22-3692 a^12 b^12 c^22-8167 a^10 b^14 c^22-2940 a^8 b^16 c^22+7493 a^6 b^18 c^22+3502 a^4 b^20 c^22-3432 a^2 b^22 c^22+2210 b^24 c^22-2912 a^22 c^24-494 a^20 b^2 c^24+3126 a^18 b^4 c^24-4692 a^16 b^6 c^24-9713 a^14 b^8 c^24-1368 a^12 b^10 c^24+9574 a^10 b^12 c^24+7018 a^8 b^14 c^24-5870 a^6 b^16 c^24-7028 a^4 b^18 c^24+3666 a^2 b^20 c^24+2210 b^22 c^24+1052 a^20 c^26-2132 a^18 b^2 c^26+54 a^16 b^4 c^26+9020 a^14 b^6 c^26+6227 a^12 b^8 c^26-9053 a^10 b^10 c^26-14546 a^8 b^12 c^26-2327 a^6 b^14 c^26+5734 a^4 b^16 c^26-2104 a^2 b^18 c^26-4250 b^20 c^26+1378 a^18 c^28+3498 a^16 b^2 c^28-2301 a^14 b^4 c^28-6475 a^12 b^6 c^28+4814 a^10 b^8 c^28+16922 a^8 b^10 c^28+9916 a^6 b^12 c^28-3550 a^4 b^14 c^28-2450 a^2 b^16 c^28+2210 b^18 c^28-1526 a^16 c^30-632 a^14 b^2 c^30+4745 a^12 b^4 c^30+1627 a^10 b^6 c^30-10446 a^8 b^8 c^30-11293 a^6 b^10 c^30+966 a^4 b^12 c^30+5552 a^2 b^14 c^30+918 b^16 c^30+177 a^14 c^32-2497 a^12 b^2 c^32-4040 a^10 b^4 c^32+2946 a^8 b^6 c^32+8756 a^6 b^8 c^32+2794 a^4 b^10 c^32-3785 a^2 b^12 c^32-1989 b^14 c^32+587 a^12 c^34+2138 a^10 b^2 c^34-23 a^8 b^4 c^34-5446 a^6 b^6 c^34-5113 a^4 b^8 c^34-76 a^2 b^10 c^34+1089 b^12 c^34-382 a^10 c^36-145 a^8 b^2 c^36+2518 a^6 b^4 c^36+4166 a^4 b^6 c^36+1961 a^2 b^8 c^36-13 b^10 c^36+14 a^8 c^38-715 a^6 b^2 c^38-1884 a^4 b^4 c^38-1532 a^2 b^6 c^38-335 b^8 c^38+90 a^6 c^40+461 a^4 b^2 c^40+597 a^2 b^4 c^40+221 b^6 c^40-48 a^4 c^42-124 a^2 b^2 c^42-73 b^4 c^42+11 a^2 c^44+13 b^2 c^44-c^46 : :

Benjamin Lee Warren and Francisco Javier García Capitán, euclid 9658.

X(72514) lies on this line: {4, 72513}

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G - Orthologic

Let ABC be a triangle, P = G = X(2) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on GC, GB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?
Orthologic center (QaQbQc, ABC) = G** = ?

For Q = X(3) = O:
Orthologic center (ABC, QaQbQc) = O* = X(36889)
Orthologic center (QaQbQc, ABC) = O** = X(1352)
Euclid 9541

Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = X(3)= O
Orthologic center (QaQbQc, ABC) = H** = ?

Q = N = X(5)
Orthologic center (ABC, QaQbqc) = N* = ?
Orthologic center (QaQbQc, ABC) = N** = ?

The locus of the orthologic center (QaQbQc, ABC) = Q**, as Q moves on the Euler line, is a line.
(OQ/OH = O**Q**/O**H**)

Locus of the orthologic center (ABC, QaQbQc) ?

H - Orthologic

Let ABC be a triangle, P = H = X(4) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on HC, HB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?
Orthologic center (QaQbQc, ABC) = G** = G of orthic = X(51)

For Q = X(3) = O:
Orthologic centers = X(4) = H

Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = X(3) = O
Orthologic center (QaQbQc, ABC) = H** = ?

For Q = N = X(5)
Orthologic center (ABC, QaQbQc) = N* = ?
Orthologic center (QaQbQc, ABC) = N** = ?

The locus of the orthologic center (QaQbQc, ABC) = Q**, as Q moves on the Euler line, is a line. (The line {4,51})
(OQ/OH = O**Q**/O**H**)

Locus of the orthologic center (ABC, QaQbQc) ?

O - Orthologic

Let ABC be a triangle, P = O = X(3) and Q a point on the Euler line.

Denote:

Bc, Cb = the orthogonal projections of B, C on OC, OB, resp.

Qa = same to Q point of the triangle ABcCb.
Similarly Qb, Qc.

ABC, QaQbQc are orthologic.

Orthologic center (QaQbQc, ABC) = Q

For Q = G = X(2)
Orthologic center (ABC, QaQbQc) = G* = ?

For Q = X(3) = O:
Orthologic center (ABC, QaQbQc) = O* = X(72422) = X(2)X(9291)∩X(4)X(290)

For Q = H = X(4)
Orthologic center (ABC, QaQbQc) = H* = ?

For Q = N = X(5)
Orthologic center (ABC, QaQbQc) = N* = ?

Locus:
The locus of the orthologic center (ABC, QaQbQc) = Q*, as Q moves on the Euler line, is a CIRCLE

LOCI

Let ABC be a triangle and P a point.

Denote:
Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

A' = the other than A intersection the circumcircles of ABC and ABcCb
Similarly B',C'

La, B, Lc = Euler lines of A'BC, B'CA, C'AB, resp.

1. Which is the locus of P such that ABC, A'B'C' are orthologic?
O lies on the locus
Orthologic center (ABC, A'B'C') = (3) = O
Orthologic center ( A'B'C', ABC) = Χ(20)

2. Which is the locus of P such that the parallels to La,Lb, Lc through A, B, C,resp, are concurrent?
O lies on the locus.
.

Circumcenters - Orthologic

Let ABC be a triangle and P a point.

Denote:

Bc, Cb = the orthogonal projections of B, C on PC, PB, resp.

Oa = the circumcenter of ABcCb.
Similarly Ob, Oc.

Which is the locus of P such that ABC, OaObOc are orthologic?

H, O, G lie on the locus.