The point D = tH = X(69) is the perspector if the inconic I(D) with centre O = X(3).
From our general theory, the inconic I(D)
(1) is the D-Ceva conjugate of T(D),
(2) intersects I(H) at X(125) - from X(525), the infinite point of T(D).
None of the other easily derived points on I(D) appear in ETC.
But I(D) has a totally diferent description. It is the locus of orthopoles of lines through
the de Longchamps point, L = X(20). It is the only inconic of this form.
X1 = X125 is the orthopole of the Euler Line, so is on I(D).
The line Io-Go (X(1)-X(7)) passes through L. It has orthopole X(1565).
Hence X2o = X(1565) is on I(D).
Note : Kimberling refers to this as an orthojoin of the point X(658) - the orthopole of the
line whose tripole is the isogonal conjugate of X(658). Why the isogonal conjugate?
Now observe that Io-Go is a weak line, while L is a strong point. Thus we have three other
lines through L, namely Ix-Gx (x = a,b,c). Each has orthopole on I(D). Call these X2a,b,c.
The centre X(305) is the isotomic conjugate of the isogonal conjugate (Conway's "retro")
of the point D.
(1) X2o, X2a, X2b, X2c lie on I(D) and on I(X(305)).
(2) X2o-X2a meets BC at the foot of the A-Cevian of X(304), and similarly for X2b, X2c.
(3) X2x (x=o,a,b,c) come as D-Ceva conjugates from intersections of T(D) with tripolars
of X(304) and its harmonics.
See Theorem 8 and Corollaries for the general version of (2).
Each line through L gives rise to a point of I(D). We should expect related lines to give
points which are somehow related.
If M and N are perpendicular lines though L, then the orthopoles are antipodes.
In other words, the orthopoles of M and N are collinear with O.
This uses lines through L which are conjugate in the Circumcircle. We might also
think of conjugacy in another conic, for example I(D) itself.
If M and N are lines through L conjugate with respect to I(D), then their orthopoles
are collinear with X(1204) (a grotesque point).
Let M be a line through L. Let Y be the pole of L with respect to I(D), and N the line Y-L.
Then the orthopoles of M and N are collinear with H.
Now a general result for inconics - see inconics part 1.
Let Y be a point on T(D), and M its polar with respect to I(D). Let Z be the meet of
M and T(D). Then the D-Ceva conjugates of Y and Z are collinear with X(69).
Now a result which uses both descriptions together. Observe that L is not on T(D).
Thus, if Y is a point on T(D), we have a line L-Y. Both Y itself and this line L-Y define
points on I(D).
Suppose Y is on T(D). Then
(1) X', the D-Ceva conjugate of Y is on I(D),
(2) X", the orthopole of the line L-Y is on I(D), and
(3) G is on X'-X".