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X(1), X(20), X(185), X(1075), X(14135), X(14709), X(14710), X(46623), X(46624), X(46625), X(46626)

excenters

other points below

Geometric properties :

K1256 is defined in page K003 where several properties are already mentioned. Recall it is the locus of M whose polar conic in K003 is a bicevian conic, actually a bicevian rectangular hyperbola since K003 is a stelloid.

Examples :

• the polar conic of X(20) is C(X2, X110), the complement of the Jerabek hyperbola.

• the polar conic of X(1) has center X(1054) and passes through X(1), X(164), X(173).

• the polar conic of X(1075) is C(X3, X15352) and passes through X(4), X(417), X(1075).

• the polar conics of X(14709) and X(14710) pass through O, K and are homothetic to the Jerabek hyperbola.

• the tangential of X(1) in K1256 is T = a (a^6-a^5 b-3 a^4 b^2+a^3 b^3+2 a^2 b^4-a^5 c-3 a^4 b c-a^3 b^2 c+2 a^2 b^3 c+2 a b^4 c+b^5 c-3 a^4 c^2-a^3 b c^2+2 a^2 b^2 c^2+a^3 c^3+2 a^2 b c^3-2 b^3 c^3+2 a^2 c^4+2 a b c^4+b c^5) : : , on the line {1,3}, SEARCH = 19.9427837484072. Its polar conic is C(X662, X40430) passing through X(1), X(21)

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K1256 meets the internal bisectors again at A1, B1, C1 and the sidelines of the excentral triangle again at A2, B2, C2. These points have barycentric coordinates :

A1 = {a^6-a^4 b^2-a^2 b^4+b^6-a^4 b c+b^5 c-a^4 c^2+6 a^2 b^2 c^2-b^4 c^2-2 b^3 c^3-a^2 c^4-b^2 c^4+b c^5+c^6,-b (-a+b-c) (a+b-c) (b+c) (-a^2+b^2+c^2),-c (-a-b+c) (a-b+c) (b+c) (-a^2+b^2+c^2)},

A2 = {a^6-a^4 b^2-a^2 b^4+b^6+a^4 b c-b^5 c-a^4 c^2+6 a^2 b^2 c^2-b^4 c^2+2 b^3 c^3-a^2 c^4-b^2 c^4-b c^5+c^6,-b (b-c) (-a+b+c) (a+b+c) (-a^2+b^2+c^2),-c (-b+c) (-a+b+c) (a+b+c) (-a^2+b^2+c^2)}, and the other points cyclically.

These triangles A1B1C1, A2B2C2 and the anticevian triangle of H are two by two perspective at Q on K1256 and on the line {20,3186}.

Q = (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^12-7 a^10 b^2-a^8 b^4+14 a^6 b^6-11 a^4 b^8+a^2 b^10+b^12-7 a^10 c^2+37 a^8 b^2 c^2-42 a^6 b^4 c^2-2 a^4 b^6 c^2+17 a^2 b^8 c^2-3 b^10 c^2-a^8 c^4-42 a^6 b^2 c^4+74 a^4 b^4 c^4-18 a^2 b^6 c^4+3 b^8 c^4+14 a^6 c^6-2 a^4 b^2 c^6-18 a^2 b^4 c^6-2 b^6 c^6-11 a^4 c^8+17 a^2 b^2 c^8+3 b^4 c^8+a^2 c^10-3 b^2 c^10+c^12) : : , SEARCH = 33.8271814510328.

The third point of K1256 on the line {20,3186,Q} is Q1 = a^14 b^2-5 a^12 b^4+9 a^10 b^6-6 a^8 b^8-a^6 b^10+3 a^4 b^12-a^2 b^14+a^14 c^2+a^12 b^2 c^2+a^10 b^4 c^2-13 a^8 b^6 c^2+15 a^6 b^8 c^2-5 a^4 b^10 c^2-a^2 b^12 c^2+b^14 c^2-5 a^12 c^4+a^10 b^2 c^4+a^8 b^4 c^4-2 a^6 b^6 c^4-13 a^4 b^8 c^4+5 a^2 b^10 c^4-3 b^12 c^4+9 a^10 c^6-13 a^8 b^2 c^6-2 a^6 b^4 c^6+14 a^4 b^6 c^6-3 a^2 b^8 c^6+3 b^10 c^6-6 a^8 c^8+15 a^6 b^2 c^8-13 a^4 b^4 c^8-3 a^2 b^6 c^8-2 b^8 c^8-a^6 c^10-5 a^4 b^2 c^10+5 a^2 b^4 c^10+3 b^6 c^10+3 a^4 c^12-a^2 b^2 c^12-3 b^4 c^12-a^2 c^14+b^2 c^14 : : , SEARCH = 4.52545557431051, also on the line {147,185}.

The third point of K1256 on the line {185,Q1} is Q2 = a^2 (a^18 b^2-3 a^16 b^4+3 a^14 b^6-3 a^12 b^8+6 a^10 b^10-6 a^8 b^12+3 a^6 b^14-3 a^4 b^16+3 a^2 b^18-b^20+a^18 c^2-a^14 b^4 c^2+6 a^12 b^6 c^2-26 a^10 b^8 c^2+34 a^8 b^10 c^2-21 a^6 b^12 c^2+14 a^4 b^14 c^2-9 a^2 b^16 c^2+2 b^18 c^2-3 a^16 c^4-a^14 b^2 c^4+14 a^12 b^4 c^4-23 a^10 b^6 c^4+25 a^8 b^8 c^4-a^6 b^10 c^4-14 a^4 b^12 c^4+9 a^2 b^14 c^4-6 b^16 c^4+3 a^14 c^6+6 a^12 b^2 c^6-23 a^10 b^4 c^6+40 a^8 b^6 c^6-37 a^6 b^8 c^6+22 a^4 b^10 c^6-11 a^2 b^12 c^6+16 b^14 c^6-3 a^12 c^8-26 a^10 b^2 c^8+25 a^8 b^4 c^8-37 a^6 b^6 c^8-6 a^4 b^8 c^8+8 a^2 b^10 c^8-17 b^12 c^8+6 a^10 c^10+34 a^8 b^2 c^10-a^6 b^4 c^10+22 a^4 b^6 c^10+8 a^2 b^8 c^10+12 b^10 c^10-6 a^8 c^12-21 a^6 b^2 c^12-14 a^4 b^4 c^12-11 a^2 b^6 c^12-17 b^8 c^12+3 a^6 c^14+14 a^4 b^2 c^14+9 a^2 b^4 c^14+16 b^6 c^14-3 a^4 c^16-9 a^2 b^2 c^16-6 b^4 c^16+3 a^2 c^18+2 b^2 c^18-c^20) : : , SEARCH = 14.8141045591615.

These points T, Q, Q1, Q2 are now X(46623), X(46624), X(46625), X(46626) in ETC (2022-01-12).

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Perspective and orthologic triangles

• A1B1C1 is perspective to the anticevian (resp. cevian) triangle of M for every M on a pK passing through {1, 4, 29} (resp. a pK passing through {1, 21, 29, 1885}).

• A2B2C2 is perspective to the anticevian (resp. cevian) triangle of M for every M on the line {1,4} or on a circum-conic (resp. on the circum-conic with perspector X1 or on a line passing through X1885).

• A1B1C1 is orthologic to the intouch triangle with orthologic centers X(1) and O1 = (-a-b+c) (a-b+c) (a^3 b-a^2 b^2+a b^3-b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2+2 b^2 c^2+a c^3-c^4) : :, SEARCH = 0.524138605561333, on the lines {1,4}, {57,1738}, {75,1088}.

• A1B1C1 is orthologic to the excentral triangle with orthologic centers X(1) and O2 = a (a^5-a^3 b^2+a^2 b^3-b^5+a^3 b c-2 a^2 b^2 c+a b^3 c-a^3 c^2-2 a^2 b c^2+2 a b^2 c^2+b^3 c^2+a^2 c^3+a b c^3+b^2 c^3-c^5) : : , SEARCH = 1.64127637225089, on the lines {8,20}, {19,2319}, {55,846}, {57,1738}.