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K1104

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

X(3), X(3224), X(32445), X(32540), X(32541), X(32542), X(32543)

A', B', C' : vertices of the ITB triangle

A1, B1, C1 : vertices of the tangential triangle

A2, B2, C2 : vertices of the tangential triangle of the ITB triangle

other points below

Geometric properties :

The ITB triangle A'B'C' is defined at K1098 and K1104 is intoduced at K410.

Recall that, for any P* on K1104, there is a pK (with respect to ABC) with isopivot P* passing through the vertices of the ITB triangle. The pivot P of this pK lies on K444 and its pole Ω lies on psK(X9233, X32, X6).

With P* = X(3), we obtain K411 = pK(X14575, X32) and with P* = X(3224), we obtain K410 = pK(X6, X194).

The polar conic (C) of X(6) in K1104 is the circum-conic of the tangential triangle A1B1C1 with center X(5023), a point on the Brocard axis. (C) also contains X(1979) and the vertices A2, B2, C2 of the tangential triangle of A'B'C'. Hence the tangents to K1104 at A1, B1, C1, A2, B2, C2 concur at X(6).

K1104 meets the line at infinity at the same points as pK(X6, S) where S = a^2 (a^4 b^4-a^2 b^6-2 a^4 b^2 c^2+a^2 b^4 c^2+b^6 c^2+a^4 c^4+a^2 b^2 c^4-b^4 c^4-a^2 c^6+b^2 c^6) : : , SEARCH = 5.93625201611584, on the lines {76, 512}, {110, 32445}, {699, 3224} and many others.

Some other centers on K1104 :

• P3 = X(32445), the isopivot of the pK with pivot X(4)

• P4, the isopivot of the pK with pivot X(14251), the tangential of X(3), on the lines {3, 76}, {6, 14251}, {32, 512}, {39, 15630}, {248, 30496}, {446, 1503}, {1634, 19120}, {1910, 2053}, {3095, 13137}, {3224, 14601}, {5025, 10342}, {17932, 19597}

• P5, the third point of K1104 on the line X(3)X(3224), on the lines {3, 3224}, {3360, 18829}, {8789, 10131}

• P6, the third point of K1104 on the line X(3)X(32445), on the lines {3, 1625, 32445}, {24, 2698}, {3148, 14251}, {3406, 8743}

• P7, the third point of K1104 on the line P3P5, the isogonal conjugate of gP7 = X(2)X(3224) /\ X(69)X(698) = anticomplement of X(3224).

coordinates :

P4 = a^2*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(a^2*b^4 - b^4*c^2 + a^2*c^4 - b^2*c^4)::

P5 = a^2*(-b^2 + a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2)*(a^8*b^4 - 2*a^8*b^2*c^2 + a^4*b^6*c^2 + a^8*c^4 + a^4*b^4*c^4 - 2*a^2*b^6*c^4 + a^4*b^2*c^6 - 2*a^2*b^4*c^6 + b^6*c^6)::

P6 = a^2*(a^2 - b*c)*(a^2 + b*c)*(a^4*b^4 - 2*a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 - 2*a^4*c^4 + b^4*c^4 + a^2*c^6)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^4*c^4 + a^2*b^2*c^4 + b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8)::

P7 = a^2*(a^6*b^4 + a^4*b^6 - 2*a^6*b^2*c^2 + a^4*b^4*c^2 - 2*a^2*b^6*c^2 + a^6*c^4 - a^4*b^2*c^4 - a^2*b^4*c^4 + b^6*c^4 - a^4*c^6 + 2*a^2*b^2*c^6 - b^4*c^6)*(a^6*b^4 - a^4*b^6 - 2*a^6*b^2*c^2 - a^4*b^4*c^2 + 2*a^2*b^6*c^2 + a^6*c^4 + a^4*b^2*c^4 - a^2*b^4*c^4 - b^6*c^4 + a^4*c^6 - 2*a^2*b^2*c^6 + b^4*c^6)::

gP7 = a^6 b^4-a^4 b^6-2 a^6 b^2 c^2+a^4 b^4 c^2+2 a^2 b^6 c^2+a^6 c^4+a^4 b^2 c^4-a^2 b^4 c^4-b^6 c^4-a^4 c^6+2 a^2 b^2 c^6-b^4 c^6::

These five points and S are now X(32540), X(32541), X(32542), X(32543), X(32548) and X(32547) in ETC (2019-05-16).

Also added are the pivots X(32544), X(32545), X(32546) on K444 associated with P5, P6, P7.

K1104a

K1104 meets the sideline BC again at U = 0 : b^2(a^2 b^2 c^2 + T) : c^2(a^2 b^2 c^2 – T) where T = (b^2-c^2)(c^2-a^2)(a^2-b^2).

V on CA and W on AB are defined cyclically.

If PaPbPc is the anticevian triangle of X(3224) then UVW and PaPbPc are orthologic at O and another rather complicated point O2.

It follows that U is the trace on BC of the perpendicular at O to PbPc.

The pK with isopivot U splits into the sideline BC and a hyperbola passing through A', B', C', A (with tangent passing through U), the A-vertex of the cevian triangle of X(6), the trace on BC of the Lemoine axis.