Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves


too complicated to be written here. Click on the link to download a text file.

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


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)


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).


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}.