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∑ cot(B – C) x (y – z)^2 = 0

X(1), X(19), X(63), X(3375), X(3376), X(3377), X(3378), X(3383), X(3384), X(3400), X(3401), X(3402), X(3403), X(3404), X(3405), X(3408), X(3409), X(21061), X(41342), X(43729)

other points described below and also in table 38.

K457 is the locus of points whose TRILINEAR coordinates are

tan(A+t) : tan(B+t) : tan(C+t) or cot(A+t) : cot(B+t) : cot(C+t), where t is any real number.

When t = 0, we obtain the Clawson point X(19) and its isogonal conjugate X(63).

K457 is a conico-pivotal isogonal cubic with singularity the incenter X(1), an isolated point on the cubic with two imaginary tangents.

For any point M on the cubic, the line through M and its isogonal conjugate M* is tangent to a fixed conic. This pivotal conic passes through X(63), X(920) and is tangent at X(63) to the line X(19)X(63). It is inscribed in the excentral triangle with perspector the intersection of the lines {1,1101} and {5,2607}, SEARCH = 13.7471524836012. It is also inscribed in the triangle bounded by the lines passing through the vertices of ABC and the corresponding traces of the trilinear polar of the root X(2617). See Special Isocubics, §8.

K457 has three real inflexion points X(19), P3 = X(3384), M3 = X(3375) obtained when t = 0, t = π/3 and t = –π/3. These points lie on the line X(19)X(2290). Their isogonal conjugates X(63), P6 = X(3376), M6 = X(3383) are the contacts of the cubic with the pivotal conic.

When t = π/6 and t = –π/6, recall that the points are P6 and M6 are the isogonal conjugates of M3 and P3. The tangents at P6, M6 to the cubic pass through M3, P3 respectively. These tangents are also tangent to the pivotal conic.

It follows that the three points X(63), P6 and M6 are the three sextactic points on the cubic. They lie on the contact conic which is the circumconic passing through X(63) and X(2964).

When t = π/4 and t = –π/4, the points are P4 = X(3378) and M4 = X(3377) on the second tangent drawn from X(19) to the pivotal cubic. The tangents at these points pass through X(63).

See Table 38 for other points related to particular values of t and their corresponding properties. See also Table 69.

***

When the coordinates tan(A+t) : tan(B+t) : tan(C+t) are barycentric coordinates, the locus is K267. Note that K457 is the barycentric product X(1) x K267.

Let T be a real number in the interval [– π/2; + π/2] and let k = tan(T).

When t varies, the locus of point ZT(t) whose TRILINEAR coordinates are : k + tan(A+t) : k + tan(B+t) : k + tan(C+t) is an acnodal circumcubic (KT) with node X(1).

All these cubics are in a same pencil generated by K457 (obtained for T = k = 0) and the union of the three internal bisectors of ABC ( obtained for T = ±π/2 hence k = ∞). These are the only isogonal cubics of the pencil since the isogonal transform of (KT) is (K–T).

For any finite k, X(1) is an isolated point and its polar conic is the same for every cubic.

Properties

• Three points ZT(t1) , ZT(t2) , ZT(t3) of (KT) are collinear if and only if t1 + t2 + t3 = – T (mod. π).

• Hence, for any t ≠ 0, the line ZT(t) ZT(–t) passes through a fixed point of (KT), namely ZT(–T) and also contains YT(t) with 1st trilinear coordinate sec(A+t) sec(A–t) or equivalently 1 / (cos 2A + cos 2t).

These points ZT(–T) and YT(t) lie on the circum-conic (C661) with perspector X(661) passing through X(1), X(10), X(19), X(37), X(65), X(75), X(82), X(91), X(158), X(225), X(267), X(596), X(759), X(775), X(876), X(897), X(921), X(969), X(994), X(1247), X(1581), X(1910), X(2153), X(2154), X(2166), X(2168), X(2186), X(2190), X(2214), X(2216), X(2217), X(2218), X(2219), X(2363), X(2588), X(2589), X(2652), X(2962), etc.

• The tangential of ZT(t) on (KT) is ZT(–T – 2t).

• (KT) has three real inflexion points obtained for t = – T/3 (mod. π/3) and the line passing through these points also contains the fixed point X(9219).

When T varies, the locus of these inflexion points is the quartic Q121.

(Kπ/4) is the locus of point Z(t) whose TRILINEAR coordinates are :

1 + tan(A+t) : 1 + tan(B+t) : 1 + tan(C+t).

• The three real (orange) inflexion points are

Z(π/4), SEARCH = 1.58923311700147,

Z(–π/12), SEARCH = -2.68691036011913,

Z(–5π/12), SEARCH = -0.437587927924940.

• (Kπ/4) contains :

Z(–π/4), SEARCH = 0.419694935342941,

Z(π/3), SEARCH = 35.3154933578967,

Z(–π/3), SEARCH = 0.309007841546565,

all lying on a line passing through X(2962),

and also the two simple points

Z(0) = 1 + tanA : : , SEARCH = 4.92522827778129,

Z(π/2) = 1 – cotA : : , SEARCH = -6.09704854108238, both on the line passing through X(92) and Z(π/4).

Recall that three points Z(t1), Z(t2), Z(t3) are collinear on (Kπ/4) if and only if t1 + t2 + t3 = – π/4 (mod. π).

It follows that two points Z(t1), Z(t2) such that t1 = t2 (mod. π/2) share the same tangential and then the tangential of Z(t) is Z(– π/4 – 2t). This is the case of Z(0), Z(π/2) with tangential Z(–π/4).

Obviously, Z(t) and Z(– π/4 – 2t) coincide when Z(t) is a point of inflexion.

Two points Z(t1), Z(t2) on (Kπ/4) such that t1 + t2 = – π/4, + π/4, π/2, 0, (mod. π) are collinear with Z(0), Z(π/2), Z(π/4), Z(– π/4) respectively.