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.
Fact 1
X1 = X125 is the orthopole of the Euler Line, so is on I(D).
Fact 2
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.
Fact 3
(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.
Fact 4
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.
Fact 5
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).
Fact 6
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.
Fact 7
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).
Fact 8
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".