1. on the dual mapping
From projective geometry, we have the dual mapping relating the inconic I(P) with the circumconic C(tP).
For a point X on C(tP), the dual of X is a line L tangent to I(P). Let Y be the point of contact of L with I(P).
For a point Y on I(P), we have a tangent L to I(P) at Y. The dual of L is a point X on C(tP). We can relate
the pairs {X,Y} using familiar operations from triangle geometry.
First of all, we observe that isotomic conjugation relates C(tP) and T(P). We now show that the operation
of Ceva conjugation (also known as Ceva quotient) relates T(P) and I(P).
Lemma 1
P-Ceva conjugation is essentially a bijection between T(P) and I(P).
For X on T(P), T(X) touches I(P) at the P-Ceva conjugate of X.
For Y on I(P), the tangent to I(P) at Y is T(X), with X the P-Ceva conjugate of Y.
Proof notes
The algebra shows at once that the P-Ceva conjugate of a point X lies on T(P)
precisely when X lies on I(P). Since P-Ceva conjugation is essentially bijective,
the result follows.
For X on I(P), it is easy to see that Y, the P-Ceva conjugate of X, is on T(P).
Maple calculation shows that the tripolar of Y touches I(P) as Y is on T(P).
Observe that X, the P-Ceva conjugate of Y is on T(Y), so is the contact point.
Finally, as P-Ceva conjugation is an involution, we get the last clause.
Theorem 2
For pairs {X,Y}, the following pairs of conditions are equivalent
(1a) X is on C(tP) and Y is the contact of I(P) with the dual of X,
(1b) X is the isotomic conjugate of the P-Ceva conjugate of Y, and
(2a) Y is on I(P) and X is the dual of the tangent to I(P) at Y.
(2b) Y is the P-Ceva conjugate of the isotomic conjugate of X.
2. some familiar operations
Above, we saw that Ceva conjugation is useful in the study of inconics as well as its more familiar role
in the theory of circumconics. Other well-known operations from triangle geometry are also useful. The
following results are trivial consequences of the definitions of the operations.
Lemma 3
The polar of X with respect to the circumconic C(P) is the tripolar of the cevapoint of P and X.
Of course, this tripolar is also the polar of P with respect to C(X).
This is related to the general notion of Hirst inversion.
Lemma 4
The pole of T(X) with respect to the circumconic C(P) is the X-Ceva conjugate of P.
Lemma 5
For X ≠ U, the Hirst inverse of X in the circumconic C(P) with pole U is the intersection of the
tripole of the cevapoint of P and X with the line U-X.
We could avoid the term "Hirst inverse", using Lemma 4 as a definition. We are particularly interested
in points on a fixed line L which are conjugate with respect to a circumconic. This can be rephrased in
terms of Hirst inversion, using a fixed point on L as the pole of inversion.
Lemma 6
The polar of X with respect to the inconic I(P) is the tripolar of the X-cross conjugate of P.
Note For X on I(P), the X-cross-conjugate of P is the P-Ceva conjugate of X.
Lemma 7
The pole of T(X) with respect to the inconic I(P) is the crosspoint of P and X.
Note For X on T(P), the crosspoint of P and X is the P-Ceva conjugate of X.
The intersection of two inconics.
Maple calculation finds four real intersections of I(P) and I(tP). Here, we describe the geometry in
the general case. It turns out that two inconics have four real intersections or none.
Theorem 8
The intersections of the inconics I(P) and I(Q) correspond to the fixed points of the isoconjugation
which interchanges P and Q. If R is a fixed point, then I(P) and I(Q) intersect the point Xo which is
the P-Ceva conjugate of Po, the intersection of T(P) and T(R),
the Q-Ceva conjugate of Qo, the intersection of T(Q) and T(R),
the intersection of T(Po) and T(Qo).
The tripolars T(Po), T(Qo) are the tangents to I(P), I(Q) at Xo.
Corollary 8.1
The inconics I(P), I(Q) have four real intersections when the barycentric product P&Q is inside
ΔABC, and none otherwise.
Note that, if R is one fixed point, then the others are the vertices of the anticevian triangle of R.
The other tripolars are the sidelines of the cevian triangle of R.
We can go a lot further.
Corollary 8.2
Suppose we choose as R a fixed point of the isoconjugation.
The other fixed points are the harmonic associates of R. Each gives a point of I(P).
Let Xo come from R, and Xa, Xb, Xc from the harmonics.
Then
Xo-Xa, BC and the A-cevian of R concur, i.e. ΔXaXbXc is perspective with the Cevian triangle of R at Xo.
Xb-Xc, BC and T(R) concur, so ΔXaXbXc and ΔABC are perspective,
AXa, BXb, CXc concur, so ΔXaXbXc and ΔABC are perspective at a point S, where
S is the crosspoint of the crosspoints of {P,R} and {R,Q},
Xo, R and S are collinear.
Geometry
Once Xo and S known, then Xa, Xb, Xc follow as simple intersections, for example
Xa is the intersection of A-S and the line joining Xo to the foot of the A-Cevian of R.
Algebra
If P = p:q:r, and R = u:v:w, then
Xo = p(q/v-r/w)2:q(r/w-p/u)2:r(p/u-q/v)2,
S = p(q/v+r/w)2:q(r/w+p/u)2:r(p/u+q/v)2.
In the case where Q is the isotomic conjugate of P, the fixed points are G and the vertices of the
antimedial triangle. The tripolars are the Line at Infinity and the sidelines of the medial triangle.
The meet of Xo-Xa with BC is then the mid-point of BC.
3. Notes on some Steve Sigur Webpages.
Steve's notation
For a point Q, ~Q denotes the dual of Q ( if Q = u:v:w, ~P is ux+vy+wz = 0 ).
tP denotes the isotomic conjugate of P.
∞ denotes the line at infinity x+y+z =0.
For points P, Q, P-Q denotes the line joining P and Q.
For lines L, M, L.M denotes the intersection of L and M.
With the point P = p:q:r, we associate several points at infinity. The first three are due to Steve.
| descriptor | coordinates |
| ~P.∞ | q-r:r-p:p-q |
| (G-P).∞ | 2p-q-r:2q-r-p:2r-p-q |
| ~tP.∞ | 1/q-1/r:1/r-1/p:1/p-1/q |
| (G-tP).∞ | 2/p-1/q-1/r:2/q-1/r-1/p:2/r-1/p-1/q |
| ~(t(~P.∞)).∞ | 1/(r-p)-1/(p-q):1/(p-q)-1/(q-r):1/(q-r)-1/(r-p) |
| (t(~P.∞)-G).∞ | 2/(q-r)-1/(r-p)-1/(p-q):2/(r-p)-1/(p-q)-1/(q-r):2/(p-q)-1/(q-r)-1/(r-p) |
The first two are really the basic ones. The others are derived by applying these operations
to the points tP and t(~P.∞).
The points of the second type account for the points X(527)-X(545) in ETC, among others.
Those of the first, second, fifth and sixth type for a given P are related in a nice way.
Some points on the Steiner Ellipse.
The isotomic conjugate of any point at infinity is on the Steiner Ellipse.
Suppose that P is any point. Then we define four points as follows :
Q = t(~P.∞)
R = t((P-G).∞)
S = t(~Q.∞)
T = t((Q-G).∞).
The pairs {Q,R}, {S,T} are antipodes on the Steiner Ellipse,
Q-S and R-T are parallel to ~P,
Q-T and R-S are parallel to P-G.
~P and P-G are conjugate directions in the ellipse.
The circumconics with perspectors Q, R pass through S, with tagents Q-S, R-S.
Note that we get the same configuration for any point on the same line through G.
We observe that taking complements gives half-sized version of the parallelogram inscribed
in the Steiner Inellipse. The vertices are Q', R', S', T'. The coordinates are simply related to
those above. For example, Q' is the barycentric square of ~P.∞.
4. Axes of the Steiner ellipses
We give a short proof of the well-nown fact that the axes of the Steiner ellipses are
parallel to the asymptotes of the Kiepert Hyperbola.
Since both Steiner ellipses have the same axes, we concentrate on the circumellipse.
We recall the affine notion of conjugate directions for a conic. If C is a conic, then two
infinite points represent C-conjugate directions if each is on the polar of the other with
respect to the conic. Provided C is not a circle, the only conjugate pair which represent
perpendicular direction are the axes. We refer to lines through G in C(G)-conjugate
directions as C(G)-conjugate diameters.
Suppose that U and V are infinite points such that tU, tV are antipodal points on C(G).
Then, if U = u:v:w, so u+v=w = 0, we have V = v-w:w-u:u-v. The line L(U) = tU-tV has
equation u(v-w)x+v(w-u)y+w(u-v)z = 0, which clearly contains G.
Fact 1.
Now we observe that T(tU) is the polar of U with respect to C(G), and that V is on T(tU).
Thus, U and V represent C(G)-conjugate directions.
We also observe that U and V are on the circumconic C which is the isotomic conjugate
of the line L(U). Since they are at infinity, they represent the asymptotic directions.
Now we see that T(tU) and T(tV) are the axes of C(G) precisely when C is rectangular.
As C already passes through G, it is rectangular only when L(U) is G-K, and then C is
the Kiepert Hyperbola.
Theorem 9
The axes of the Steiner ellipses are parallel to the asymptotes of the Kiepert Hyperbola.
They are the tripolars of the intersections of the Steiner Ellipse with the line G-K.
Observation
The axes are the bisectors of the lines G-K and G-X(99). The latter also has X(115) and
X(671) and X(543). I do not see a quick proof of this.
Equations for the axes.
Suppose that we take a general point P = p:q:r (other than G). Then the inverse of P in
C(G) is P^ = p2-qr:q2-rp:r2-pq. By its nature, it is on the line G-P. Say this line meets
C(G) at X and Y. Then (P,P^,X,Y) is harmonic, by general projective theory. The we can
write X = P^+kP, Y = P^-kP. Straight-forward calculation shows that X, Y are on C(G)
when k2 = p2+q2+r2-pq-qr-rp. Then T(X), T(Y) are the corresponding C(G)-conjugate
diameters. We have proved a further result :
Theorem 10
For P ≠ G, the C(G)-conjugate diameters derived from G-P are T(X), T(Y) with X,Y on
G-P and C(G). We also have
(1) tX on T(Y) and tY on T(X),
(2) The barycentric product of tX and tY is q-r:r-p:p-q, where P = p:q:r.
(3) The constant k2 is the sum of the coordinates of P^.
Part (1) is proved as Fact 1 above. It does not depend on identifying X and Y.
Part (2) is a simple consequence of the equation for L(U) above.
Notes
It is clear that the points X, Y depend on the line through G, not on the choice of P on the line.
The value of k2 is independent of the choice of P. [To see this, replace p:q:r by p+t:q+t:r+t].
Now take P = K (or any point on G-K). Then T(X), T(Y) are the axes of the Steiner ellipses.
The infinite points of the axes are E530, E531 (Brisse notation), not to be confused with
X530, X531(Kimberling notation). Different choices of P give different looking descriptions.
Here k2 = a4+b4+c4-a2b2-b2c2-c2a2.
P = K = X(6), then P^ = X(385) (= a4-b2c2: .. : .. )
X = X(385) + kX(6), tX = X(804) - kX(512) (X(804) = (b2-c2)( a4-b2c2) : .. : .. )
We could equally write tX as the barycentric product of X(523) and Y = X(385) - kX(6).
We note that X(804) is the barycentric product of X(523) and X(385).
P = X(524), the infinite point on G-K. X(524) = 2a2-b2-c2 : .. : .. .
The geometry tells us that P^ is G, but the algebra gives it as 3k2:3k2:3k2.
In our formula, "k" should be replaced by 3k as the root of the sum of the coordinates of P^.
Then X = 3k2G + 3kX(524). This simplifies as X(524) + kG.
It follows that tX = X(690) - kX(523). X(690) is the barycentric product of X(523) and X(524).
Gibert has also given tX as X(543) - kX(524). This seems to come from a point P not listed in
the current ETC (April 2006). It is the barycentric product of X(524) and X(99). It is the
intersection of G-K and X(99)-X(110); the latter contains X(690).
The foci of the ellipses
We begin with those of the inellipse. Those of the circumellipse are their anticomplements.
The foci of the inellipse :
(1) lie on an axis,
(2) are reflections in G,
(3) are isogonal conjugates.
For those on the axis with direction tX, the foci must be of the form tX+sG, tX-sG for some s.
This follows from conditions (1) and (2). For (3), the barycentric product of the foci must be K.
This leads to an equation for s2, so we have the foci.
Using the fact that tX is at infinity, the anticomplements are 2tX-sG and 2tX+sG.
Steve Sigur's Foci of the Steiner Conics gives the formulae which arise from P = K (due
to Peter Moses) and those which arise from Bernard Gibert's description mentioned above.
5. Points on an inconic derived from a general point.
In Simmons Conics,
Steve defines points on inconics by algebra. The appear as the
barycentric products of P with points on the Steiner Inellipse. Here, we show how some
of these can be described geometrically as intersections with other inconics.
Suppose that we have an inconic I(P), with P = p:q:r, and a point U = u:v:w ≠ G.
Let Q be the isotomic conjugate of the image of P under the isoconjugation fixing U.
Thus Q = p/u2:q/v2:r/w2. If P and U are triangle centres, then I(P) and I(Q) meet
in one triangle centre and three other points. See Theorem 8. The triangle centre is
the point X1 = p(v-w)2:q(w-u)2:r(u-v)2.
X1 can be constructed as the P-Ceva conjugate of the point Y1 = p(v-w):q(w-u):r(u-v).
Y1 is the intersection of T(P) and T(R), where R is the barycentric product of P and tU.
As in Construction 6 of inconics page, we can derive three further points on I(P).
The webpage shows how these can be constructed with a straight-edge. In terms of
coordinates, we get the Yi as the barycentric products of P with the infinite points in
section 3 to get Q,R,S,T, and the Xi as barycentric products of P with their squares.
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Notation
Fn, Fs denote the Fermat (Isogonic) Points X(13), X(14) in ETC,
In, Is denote the isodynamic Points X(15), X(16).
Definition The Simmon's Conics are I(Fn) and I(Fs).
| Conic | Perspector/Focus | Second Focus | Centre |
| I(Fn) | Fn | In | X396 |
| I(Fs) | Fs | Is | X395 |