Inconic intersections and desmic structures

Let P be the point p : q : r, and Ro the point u : v: w.
The harmonics of Ro are Ra = -u : v : w, Rb = u :-v : w, Rc = u : v :-w.
The vertices of the Cevian Triangle of Ro are Ra, R,b Rc.
Let Q be the R2-isoconjugate of P, so Q = u2/p : v2/q : w2/r.

(1) The inconics I(P) and I(Q) with perspectors P and Q intersect in four real
points.These are Xo, Xa, Xb, Xc. The barycentric coordinates are given below.

The points Xo, Xa, Xb, Xc are easily constructed by ruler alone :

(2) The point Xo is the intersection of the tripolars of P1 and Q1, where
P1 is the intersection of T(P) and T(Ro),
Q1 is the intersection of TRo) and T(Q).
T(P1) is the tangent to I(P) at Xo,
T(Q1) is the tangent to I(Q) at Xo.
Xa, Xb, Xc can be obtained by replacing Ro by Ra, Rb, Rc respectively.
We will give an easier construction of Xa, Xb, Xc shortly.

As is clear from the barycentrics below, any three of Xo, Xa, Xb, Xc are the
vertices of a triangle perspective with ΔABC. The perspectors are denoted
by Yo, Ya, Yb, Yc.

The points Yo, Ya, Yb, Yc can also be constructed as follows:

(3) Yo is the crosspoint of the crosspoints of P and Ro and of Ro and Q.
Ya, Yb, Yc can be obtained by replacing Ro by Ra, Rb, Rc respectively.
Again, there are easier ways to construct Ya, Yb, Yc. These appear below.

It is easy to verify from the coordinates that Ro, Xo and Yo are collinear.
Let S be the harmonic of Ro with respect to Xo and Yo.
It is also the harmonic of Rn with respect to Xn, Yn (n = a,b,c).

Constructions
Let us assume that we have constructed Xo and Yo as above. We then have
much simpler construction of the other points.

Xa is the intersection of XoRa and AYo. Xb, Xc can be similarly described.
Ya is the intersection of BXc and CXb. Yb, Yc can be similarly described.
Alternatively, Ya is the intersection of RaXa and AXo, and so on.
S is the intersection of any two of XoYo, XaYa, XbYb and XcYc.

The results for Yn are just the construction of the desmic mate of ΔXaXbXc.

(4) The points above belong to a desmic structure :

 S A B C Xo Xa Xb Xc Yo Ya Yb Yc Ro Ra Rb Rc

Naturally, we can permute the suffices {o,a,b,c} in pairs to get alternative ways
to describe the desmic structure.

Of course, each such desmic structure (involving A, B, C) is associated with
two isocubics, one pivotal, the other non-pivotal. For the first of the structures
above, these are pK(T,S) and nK(T,Ro,Xo), where the common pole T is the
barycentric product of Xo and Yo.

Barycentric coordinates

general form for Xo, Xa, Xb, Xc, Yo, YA, Yb, Yc
p(q/v+αr/w)2 : q(r/w+βp/u)2 : r(p/u+γq/v)2.

 α β γ Xo -1 -1 -1 Xa -1 +1 +1 Xb +1 -1 +1 Xc +1 +1 -1 Yo +1 +1 +1 Ya +1 -1 -1 Yb -1 +1 -1 Yc -1 -1 +1

S = p((q/v)2+(r/w)2) : q((r/w)2+(p/u)2) : r((p/u)2+(q/v)2).

T = p2((q/v)2-(r/w)2)2 : q2((r/w)2-(p/u)2)2 : r2((p/u)2-(q/v)2)2.

Example P = K = X(6), Ro = I = X(1).
Here, we are looking at the intersections of I(K), the Brocard Inellipse,
with I(G), the Steiner Inellipse.
Q = G = X(2),
Xo = X(1015),
Yo = X(1500),
S = X(39),
T = X(1084).
The tangent to I(K) at Xo is T(X(649)),
The tangent to I(G) at Xo is T(X(513)).
Note that the ETC entry for X(1500) indicates close relations with X(1015) and X(39),
and the collinearity of X(1), X(39), X(1015) and X(1500).

Example P = G = X(7), Ro = G = X(2).
Here, we are looking at the intersections of I(G), the incircle,
with I(N), the Mandart Inellipse.
Q = N = X(8),
Xo = X(11),
Yo = X(55),
S = X(497),
T not listed in ETC.
The tangent to I(G) at X(11) is T(X(514)).
The tangent to I(N) at X(11) is T(X(519)).

Appendix
Some consequences of the existence of the desmic structure.

(5) We have triangle perspectivities as follows :

ΔXaXbXc is perspective with the Cevian triangle of Ro at Xo,
ΔXoXcXb is perspective with the Cevian triangle of Ra at Xa,
ΔXcXoXa is perspective with the Cevian triangle of Rb at Xb,
ΔXbXaXo is perspective with the Cevian triangle of Rc at Xc,
ΔXaXbXc is perspective with ΔABC at Yo,
ΔXoXcXb is perspective with ΔABC at Ya,
ΔXcXoXa is perspective with ΔABC at Yb,
ΔXbXaXo is perspective with ΔABC at Yc,
ΔYaYbYc is perspective with ΔABC at Xo,
ΔYoYcYb is perspective with ΔABC at Xa,
ΔYcYoYa is perspective with ΔABC at Xb,
ΔYbYaYo is perspective with ΔABC at Xc.

(6) XoYo, XaYa, XbYb, XcYc concur at S.

(7) We have the following harmonic ranges (and implicit collinearities) :
(Xo, Yo, Ro, S)
(Xa, Ya, Ra, S)
(Xb, Yb, Rb, S)
(Xc, Yc, Rc, S).