[Le Viet An]:
Let ABC be a triangle.
Denote:
A'B'C' = the circumcevian triangle of G.
Oa = (Perpendicular bisector of GA') /\ (Perpendicular from G to BC)
ie Oa is the center of the circle passing through A' and G and GOa is perpendicular to BC
Similarly Ob, Oc
Then
1. AOa, BOb, COc are concurrent at a point D..
Point?
2. The six circumcenters of DAOb, DAOc, DBOc, DBOa, DCOa, DCOb are concyclic.
Center of the circle?
3. Let (O') be the circle tangent internally to circles (Oa), (Ob), (Oc).
Then (O), (O') are tangent.
Touchpoint?
1) X(14494)
2)
O* = 1584*S^8+48*(3*SA^2-21*SB*SC+ 20*SW^2)*S^6-(96*SA^2+336*SB* SC-221*SW^2)*SW^2*S^4+(7*SA^2+ 47*SB*SC-2*SW^2)*SW^4*S^2-SB* SC*SW^6 : : (barys)
= on lines {}
= [ 2.6847969914825690, 1.8548837292193080, 1.1173771348405960 ]
3) X(105)
------------------------------ -----------
More:
(O’) has center:
O’ = X(1)X(2) ∩ X(3)X(105)
= 3*a^3-(b+c)*a^2+(b^2-6*b*c+c^ 2)*a+(b^2-c^2)*(b-c) : : (barys)
= on lines: {1, 2}, {3, 105}, {7, 238}, {9, 4310}, {31, 9776}, {56, 948}, {120, 3813}, {142, 4307}, {218, 3487}, {241, 7288}, {244, 5744}, {279, 1447}, {294, 7124}, {329, 748}, {390, 1738}, {443, 4339}, {516, 4859}, {527, 15601}, {537, 15590}, {726, 3161}, {740, 4402}, {885, 905}, {896, 2094}, {962, 9441}, {982, 5273}, {1001, 4000}, {1086, 5698}, {1212, 2275}, {1279, 2550}, {1386, 4648}, {1449, 4989}, {1475, 3333}, {1743, 5542}, {2263, 8732}, {3246, 5880}, {3361, 3598}, {3475, 4383}, {3485, 5228}, {3523, 11512}, {3576, 7390}, {3731, 4353}, {3751, 11038}, {3973, 5850}, {4220, 8273}, {4327, 8232}, {4419, 15254}, {4869, 5847}, {4966, 5839}, {5129, 13161}, {5247, 11037}, {5255, 11024}, {5573, 5745}, {5731, 7385}, {6361, 13635}, {6666, 7174}, {7407, 8227}, {8056, 10164}, {8616, 9778}, {10165, 11200}
= reflection of X(7613) in X(4859)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (142, 7290, 4307), (3616, 5222, 1)
= [ 11.0222200720844000, 2.3124359469655250, -3.0474312069537860 ]
and radius r’ = S*SW/(2*s*|3*s^2-4*SW|)
------------------------------ -----------
(Oa) has center
Oa = 3*S^2+SA^2-SB*SC-SW^2 : SW*SC+2*S^2 : SB*SW+2*S^2 (barys)
and radius ra = a*|cot(ω)|/6, where ω = Brocard angle of ABC
The triangle OaObOc is perspective to these triangles at the given centers (centers=0 are non ETC-centers to be checked later):
(ABC, 14494), (anti-Artzt*, 2), (1st anti-Brocard, 0), (anti-McCay $-, 0), (anticomplementary, 0), (Artzt*, 2), (1st Brocard, 0), (McCay $-, 0), (medial, 114), (inner-Napoleon, 0), (outer-Napoleon, 0), (1st Neuberg, 0), (2nd Neuberg, 0), (1st tri-squares*, 0), (2nd tri-squares*, 0), (inner-Vecten, 0), (outer-Vecten, 0)
Notes: * means homothetic triangles and $- means inversely similar triangles.
OaObOc and triangles in the following list are orthologic with the given centers:
(ABC, 2, 2), (ABC-X3 reflections, 2, 376), (anti-Aquila, 2, 551), (anti-Ara, 2, 428), (anti-Artzt, 11184, 1992), (1st anti-Brocard, 114, 7840), (4th anti-Brocard, 0, 9870), (5th anti-Brocard, 2, 12150), (6th anti-Brocard, 114, 12151), (anti-Euler, 2, 376), (anti-Mandart-incircle, 2, 4421), (anti-McCay, 0, 385), (anticomplementary, 2, 2), (Aquila, 2, 3679), (Ara, 2, 9909), (Artzt, 11184, 9770), (1st Auriga, 2, 11207), (2nd Auriga, 2, 11208), (1st Brocard, 114, 599), (4th Brocard, 0, 2), (5th Brocard, 2, 7811), (6th Brocard, 114, 9939), (circummedial, 5, 2), (Euler, 2, 381), (5th Euler, 5, 2), (outer-Garcia, 2, 3679), (Gossard, 2, 1651), (inner-Grebe, 2, 5861), (outer-Grebe, 2, 5860), (Johnson, 2, 381), (inner-Johnson, 2, 11235), (outer-Johnson, 2, 11236), (1st Johnson-Yff, 2, 11237), (2nd Johnson-Yff, 2, 11238), (Lucas homothetic, 2, 12152), (Lucas(-1) homothetic, 2, 12153), (Mandart-incircle, 2, 3058), (McCay, 0, 7610), (medial, 2, 2), (5th mixtilinear, 2, 3241), (inner-Napoleon, 0, 9761), (outer-Napoleon, 0, 9763), (1st Neuberg, 0, 8667), (2nd Neuberg, 0, 9766), (3rd Parry, 0, 2), (1st tri-squares-central, 0, 13663), (2nd tri-squares-central, 0, 13783), (3rd tri-squares-central, 2, 13846), (4th tri-squares-central, 2, 13847), (1st tri-squares, 11184, 13639), (2nd tri-squares, 11184, 13759), (3rd tri-squares, 0, 2), (4th tri-squares, 0, 2), (inner-Vecten, 0, 591), (outer-Vecten, 0, 1991), (X3-ABC reflections, 2, 381), (inner-Yff, 2, 10056), (outer-Yff, 2, 10072), (inner-Yff tangents, 2, 11239), (outer-Yff tangents, 2, 11240)
OaObOc and triangles in the following list are parallelogic with the given centers:
(4th anti-Brocard, 0, 13168), (anti-McCay, 6055, 8597), (4th Brocard, 0, 4), (McCay, 6055, 381), (1st Parry, 2, 9123), (2nd Parry, 2, 9185), (3rd Parry, 0, 9147)
OaObOc is (respective centers are given):
· Inversely similar to these triangles: (4th anti-Brocard, 0), (anti-McCay, 2), (4th Brocard, 0), (McCay, 2), (3rd Parry, 0)
· directly similar to these triangles: (circumsymmedial, 0)
· directly homothetic to these triangles: (anti-Artzt, 2), (Artzt, 2), (1st tri-squares, 0), (2nd tri-squares, 0)
------------------------------ -
Let At be the touch point of (O’) and (Oa) and similarly Bt, Ct
At = a^3-3*(b-c)^2*a+2*(b^2-c^2)*( b-c) :
b^3-(c+4*a)*b^2-(3*c^2-6*c*a+ a^2)*b+(3*c-a)*(c-a)*c :
c^3-(b+4*a)*c^2-(3*b^2-6*b*a+ a^2)*c+(3*b-a)*(b-a)*b (barys)
AtBtCt is perspective to: (inner-Conway, 2), (intouch, 2), (OaObOc, O’)
César Lozada
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