Notation
We will work in barycentric coordinates throughout.
The triangle centre [f(a,b,c],f(b,c,a),f(c,a,b)] is abbreviated to <f(a,b,c)>.
We use the numbering of ETC, so X(2) = <a2>.
The Parry Circle
The Parry Circle P passes through X(2),X(15),X(16),X(23),X(110),X(111),X(352),X(353).
The centre of the Parry Circle is X(351).
P is orthogonal to the CircumCircle C : a2yz+b2zx+c2xy = 0,
and
also to the Brocard Circle B : b2c2x2+c2a2y2+a2c2z2 = a4yz+b4zx+c4xy.
This has diameter X(3)X(6). The equation may be rewritten in the form
(x+y+z)(b2c2x+c2a2y+c2z) = (a2+b2+c2) (a2yz+b2zx+c2xy).
Central Circles
In general, any circle other than C has an equation of the form
(*) (x+y+z)(x/f+y/g+z/h) = k(a2yz+b2zx+c2xy).
If [f,g,h] is a triangle centre, and k symmetric in a,b,c, then we have a central circle.
Note that this differs from Kimberling's definition of centre function, not only because it
is in barycentrics, but we put f,g,h as denominators. Our [f,g,h] is the isogonal conjugate
of Kimberling's point. We believe that this is more meaningful, in our version, B has X(6)
as its centre function. This is the tripole of the second factor on the left of (*). This is
the Lemoine Axis.
For P, the centre function is <b2c2(b2-c2)(b2+c2-2a2)> = X(691), which is on C.
Also k = (b2-c2)(c2-a2)(a2-b2)/3.
Note that the centre X(351) = <a2(b2-c2)(b2+c2-2a2)> is on the Lemoine Axis.
The Tripolar Centroid
This notion is due to Darij Grinberg and defined in ETC (see X(1635)).
For U = <u>, TG(U) = <u(v-w)(v+w-2u)>.
For a discussion of this in relation to conics, see Hyacinthos 8007-11.
Here, we note that TG(X(6)) = X(351), the centre of P, and generalize this.
The Parry Conic
For the point U, we write
UC for the circumconic uyz+vzx+wxy = 0. This has perspector U.
UB for the conic, vwx2+wuy2+uvz2 = u2yz+v2zx+w2xy. and
UP for the conic 3(x+y+z)(x/f+y/g+z/h)=(v-w)(w-u)(u-v)(uyz+vzx+wxy),
where [f,g,h] is the isotomic conjugate of <vw(v-w)(v+w-2u)>.
To avoid degeneracy of UP, we insist that U does not lie on a median.
Since we will work with the centre of UC, we insist that U is not on the Steiner Inellipse.
In any event, U on the Steiner Inellipse gives a degenerate UP as a parallel line pair.
Of course, these are obtained by replacing a2,b2,c2 by u,v,w in all the
definitions related
to the Parry Circle. Purely by algebra, since a,b,c satisfy no algebraic relation we have
the useful fact that, if a point X lies on a line or conic related to P, then this replacement
will give a point X' on the corresponding object for UP. Naturally, this extends to any
concept defined by incidence, such as intersection points and tangency. Rather less obvious
is the fact that ratios can also be translated. For example, the mid-point of <u> and <u'>
is the point <u(u'+v'+w')+u'(u+v+w)>, so can be successfully translated. In the general
case of ratio k, we must also translate the ratio. It follows that the centre of a conic and
homothety correspond. The former can also be seen from the fact that the centre of a
conic is the pole of the line at infinity, and can be obtained algebraically from the equation.
In the same way poles and polars, both trilinear and with respect to a conic, correspond.
We have many immediate consequences. Some of these can be seen easily from the
equations for UC,UB and UP. We write Un for the analogue of X(n).
UC has centre U3 = <u(v+w-u)>, the X(2)-Ceva conjugate of U.
UC passes through U110 = <u/(v-w)> and U111 = <u/(v+w-2u)> and also
U691 = <u/(v-w)(v+w-2u)>. U110 and U111 lie on the tripolar of U691, as does U.
UB has centre U182 = <u(u2-uv-uw-2vw)>.
UB passes through U and U3. The line UU3 is a diameter, i.e. passes through U182.
UP has centre U351 = TG(U).
UP passes through X(2), U110, U111, and also the points
U23 = <u(v2+w2-u2 -vw)>.
U352 = <u(u2+v2+w2+5vw-4uv-4uw)>.
U353 = <u(4u2-2v2-2w2-vw-4uv-4uw)>.
The analogues of X(15) and X(16) are rather complicated, involving square roots.
U3,U352,U353 are collinear, as are X(2),U and U353.
As circles, C,P and B are pairwise homothetic. It follows that UC,UP
and UB are pairwise
homothetic.
They therefore have parallel axes, and equal aspect ratios. The conics are ellipses
if U is inside the steiner Inellipse, and hyperbolas if it is outside.
Orthogonal Circles
We wish to generalize the concept of orthogonal circles.
Suppose that C is the centre of the conic C and that X is any point. The C-inverse
is
the
intersection of CX with the polar of X with resepct to C. For a circle C,
this is the inverse
of X with respect to C. For a curve K, the C-inverse of K
is the set of
C-inverses of the points of K.
Two circles A and A' are orthogonal if any one of the following holds.
(1) A,A' meet at P and the tangent at P to A passes through the centre of A'.
(2) the A-inverse of A' is A'.
(3) A,A' meet at P, and the tangents at P are perpendicular.
As they stand, (1),(2) are asymmetric, but it easy to prove that we can reverse the roles
of A and A'.
We shall say that conics D and E are mutually inverse if (1) and (2) hold.
From the results for the Parry Circle, we have the following information :
UP and UC are mutually inverse,
UP and UB are mutually inverse.
Results about UC,UB,UP
We add an additional useful point U187 = <u(u-2v-2w)>.
Less important, we also have
U99 = <1/(v-w)>, the intersection of UC and the Steiner Ellipse.
U574 = <u(u-2v-2w)>, the inverse of U187 for UB.
U892 = <u/f>, where <f> is U351.
We have the following collinearities :
U,U3,U15,U16,U182,U187, the tripolar of U110,
U,U23,U353,
U,U110,U111, the tripolar of U691,
U,X(2),U352, the tripolar of U99.
X(2),U3,U23,
X(2),U110,U182,
X(2),U99,U111, the tripolar of U892,
U3,U352,U353,
U23,U111,U187,U691,
U110,U187,U352,
U111,U182,U353.
U187,U351 lie on the tripolar of U.
U182 is the mid-point of (U,U3).
U187 is the mid-point of (U15,U16),(U23,U691).
We also have the following pairs of inverses:
UC-inverses (U,U187),(X(2),U23),(U15,U16),(U352,U353),
UB-inverses (X(2),U110),(U15,U16),(U111,U353).
All but the first are pairs of points on UP.
Construction
The various points and objects can be constructed provided we have conic tools such as
those provided by Cabri. We may proceed as follows.
(1) U3 as the X(2)-Ceva conjugate of U.
(2) UC as the conic with centre U3 through the vertices A,B,C.
(3) U187 as the UC-inverse of U.
(4) U23 as the UC-inverse of X(2).
(5) U691 as the reflection of U23 in U187 - it lies on UC.
(6) U111 as the intersection of the tripolar of U691 and U23,U187.
(7) U351 as the tripolar centroid of U - see below.
(8) UP as the conic with centre U351, through U2,U23,U111.
(9) U182 as the mid-point of U,U3.
(10) U110 as the intersection of the tripolar of U691 and X(2),U182
(11) V,W as the intersection of UP and the polar of U182 with respect to UP.
(12) UB as the conic with centre U182, through U,V,W.
(13) U353 as the UB-inverse of U111.
(14) U352 as the UC-inverse of U353.
(15) The pair (U15,U16) as the intersections of UP with the line U,U3.
Notes
The Tripolar centroid can be constructed as in ETC, but it may be easier to observe that
the coordinates reveal it as the intersection of the tripolars of U and U' = <u(v-w)>, the
point at infinity on the tripolar of U.
The construction has been implemented in Cabri.
Seven concurrent lines.
This was observed from the Cabri sketch, we have not verified all the algebra.
First we note that the conics UC,UP meet at U110,U111, so we have the radical axis,
L1, which is also the polar of U3 for UP and the polar of U351 for UC.
The intersections of UB and UP are unnamed, but define the line L2 which is also
the polar of U351 for UB and of U182 for UP.
From the equations, it makes sense to define the radical axis of UC and UB as the line
L3 through U187,U351 - the analogue of the Lemoine Axis.
We have three lines which turn out to be concurrent. In the case of the Parry Circle,
this is the radical centre of the circles. Thus concurrence holds in general.
We can add the lines L4 through U23,U352 and L5 through X(2),U353.
Finally, we have the tangents L6, L7 to UP at U15 and U16.
Note that L1 and L3 are the tripolars of U691 and U respectively.
The point of concurrence is <u(v2+w2-2u2-4vw+2uv+2uw)>.
A less prolific configuration is that the following lines are concurrent :
X(2),U99,U111,
U,U3,U15.U16,....
U110,U353,
the line L2 as above,
polar of U187 for UB,
The point of concurence is U574.