The incentral circumconics

Contents The incentral circumconic Properties Construction of axes Points The projective map Point table Locus definition Extraversions Perspectors and Centers updated: 10-20-06

The incentral circumconic is the isotomic conjugate of the dual of I_o, the incenter, with perspector I_o. It is a weak conic, coming in four versions, one each generated by ~I_x (x = o,a,b,c), the dual lines of the 4 incenters.

Floor van Lamoen calls this the "billiard conic" since the triangle ABC follows a billiard path. This is true of all 4 conics.

These conics are weakened versions of the circumcircle. We present the I_o circumconic as an intermediate case between the Steiner ellipse and the circumcircle.

Notation is here. Generating circumconics as the isotomic conjugate of a line is part of the affine theory of triangle conics.

Figure: The I_o-circumconic showing the conic, perspector I_o, center M_o, its axes and some points on it. The notation 162_o indicates the original version (of 4) for X162, found in the Encyclopedia of Triangle Centers (ETC). The other versions of this point are each on other versions of the conic. The conic is tangent to the exCevian lines of I_o at the vertices. The axes are light green lines and are parallel to the asymptotes of the Feuerbach hyperbola.

The "weakened" operation

If the coordinate representation of an object is a function of the squares of the edgelenghs, the substitution b² –> b, etc,. is known as "weakening" the equation. This operation is outside most traditional triangle geometry and preserves affine and projective properties. It is not usually one to one. Its inverse is called "strengthening."

Properties of the 4 I_x-circumconics

1. The perspector of the central version is I_o , which being inside the Steiner ellipse, means it is always an ellipse. The excenters are the perspectors of the other 3, and being outside the Steiner ellipse, are always hyperbolae. The central equation is

a/x + b/y + c/z = 0

This conic is the weakened form of the circumcircle and, as such, shares many of its properties. Since the incenter is always inside the Steiner inellipse, this conic is always an ellipse. The perspector I_o has the special algebraic property that its extraversions are the same as its ex-versions, often called harmonic associates. This fact is important below.

2. The four centers are the Mittenpunkts M_x = mtd I_x (x = o,a,b,c).

The Mittenpunkt and its extraversions are desmic. The I_x—M_x lines concur at K.

3. Its tangents at the vertices are the ex-Cevian lines of I_x and analogously for the other versions. The Cevian lines of the I_x are the angle bisectors, two per vertex, making 6, which meet, two at a time, at the triangle edges and 3 at a time at the 4 incenters. The effect of this is that these conics are tangent to eachother, two at each vertex.

Proof: ayz + bzx + cxy = z(ay+bx) + cxy = 0. Since the last term intersects the conic at C twice and z = 0 does not, the factor ay+bx must have a double intersection and hence be the tangent. This is an ex-Cevian line of I_o.

4. The axes are parallel to those of the Kiepert hyperbola; also the M_o circumconic and M_o inconic.

4. Its dual is the tI_o inconic, an ellipse with center S_o = mI_o.

5. Its switched conics, an ellipse and three hyperbolas, switches the center and perspector, and has perspector M_o and center I_o. Its equation is

as_a/x + bs_b/y + cs_c/z = 0.

It is generated as the isotomic conjugate of the dual of the Mittenpunkt.

The axes of circumconics whose perspectors are on a line through K are parallel. Hence the axes of the M_x and I_x circumconics are parallel.

6. Its fourth intersection with the Steiner ellipse is wS = 190_o = : 1/(c–a) :, a weakened Steiner point, and with the circumcircle wF = 100_o = : b/(c–a) :, a weakened Focus point. They have no fourth intesections with eachother, since each is tangent to one of the others at a vertex, making a double intersection there.

Since the incenter, the mittenpunkt, and the symmedian point are colinear, their duals meet at the same point. In conics-world this means that the I_o circumconic, the M_o circumconic, and the circumcircle (the K circumconic) meet at the same point as seen in the following picture.

Construction of the axes

 

Points on I_o circumconic

Peter Moses has found the ETC points on the original version of this conic. All are weak points except for the last two, which are related to the Euler intersections with the circumcircle. This was a pleasing case where the ETC was almost all information with little noise. It is also a case where the discovered points coincide well with the points the affine theory generates. Point number in red are weak points (almost all of them), green indices fissile or 8-fold. Lightened points are ones that I think should be ignored. I have placed the second barycentric coordinate after each X-number.

[PM] incentral circumconic, center 9, perspector 1
{88 b/(c+a–2b), 100 b/(c–a), 162 b/(c²–a²)S_B, 190 1/(c–a) , 651 b/(c–a)s_b, 653 1/(c–a)s_bS_B, 655 1/(c–a)s_b(4S_BB–c²a²) , 658 1/(c–a)s_bb, 660 b/(c–a)(b²–ca), 662 b/(c²–a²), 673 1/(c²+a²–ab–bc), 771 , 799 1/b(c²–a²), 823 1/b(c²–a²)S_BB, 897 b/(c²+a²–2b²) , 1156, 1492 b/(c–a)(c²+a²+ac),1821 1/bS_B^2– , 2349 ,2580,2581}

[PM again] let J = OH / R then
X(2580) = a (J–1) S_A + 2 S_BC /a :: (= X(1113)/a)
X(2581) = a (J + 1) SA – 2 SB SC/a :: (= X(1114)/a)

These points surprisingly turn out to be easy to understand, so here is a little tutorial on centers on a conic curve.

Step 1: We begin with the centers on the line at infinity. These divide into groups, originating with the famous triangle centers. The incenter has its points at infinity, the orthocenter has its points at infinity, and so on. These groups are colored in the chart below.

Step 2: The isotomic conjugate of a point on the line at infinity is on the Steiner circumellipse. These points divide into groups of points associated with triangle centers just as the line at infinity did. The relation of the line at infinity to the Steiner ellipse is grounded in affine invariance.

Step 3: Use the perspector P of a second circumconic to implement the projective map ABCG –> ABCP. Algebraically this map is equivalent to barycentric multiplication by P. Geometrically its implementation is given below. Points on the new circumconic are thus divided into groups associated with triangle centers just as the line at infinity and the Steiner ellipse were.

The view that the structure of the triangle itself is reflected in the distribution of points on a conic is a very heady and pleasing one.

Points originate on the line at infinity and percolate down to the conics through the mechanism of the Steiner ellipse. Hence all conics have contributions from all centers in the plane of the triangle. These families are shown as colors on the following table.

The geometry of this projective mapping.

A point on the line at infinity can be mapped to the Steiner ellipse by the isotomic conjugate. This point can be mapped to the I_o-circumellipse using this projective map :y: –> :by: . This same map takes the I_o-circumconic to the circumcircle. We call this the I_o projective map, since it takes ABCG to ABCI_o.

The fourth intersection of two conics allows us to easily implement the mapping of a point from one conic to the other. Simply draw a line from the 4th intersetion through the desired point. The second intersection with the other conic is the projected point. This picture shows the projection from the G-circumconic, the Steiner ellipse, to the I_o-circumconic (and vice-versa) and the circumcircle. Numbers like 673_o refer to the original version (of 4) of the quartile point X673. There are two projection centers 190_o for the Steiner to I_o projection and 100_o for the I_o to circumcircle one.

The following table shows how easy and effective this point generation process is. Each box lists a point, giving its second coordinate and the ETC X-number (if known) in red. Below it is the name of the point and/or the affine invarient constructive notation. The green section represents points that are derived from I_o. The orange section, those from the Gergonne point G_o. Blue from the symmedian point and purple from the orthocenter.

What is demonstrated here is the easy relation between points on infinity and these three conics and among the conics themselves. There is an equal number of significant points on each. We can also see how various points of the geometry plane contribute to points on the objects of the plane.

The I_o-circumconic can be seen to be an intermediate case between the Steiner ellipse and the circumcircle. The motion of a point from Steiner ellipse, to I_o-Circumconic, to circumcircle is what I have elsewhere termed an "orbit."

The red mark at the right of a row indicates a point not in the above list of Peter's from ETC.

line at infinity	Steiner ellipse	Io-Circumconic	Circumcircle
(:m:)	(:1/m:)	(:b/m:)	(:b2/m:)
	From the Incenter
514 c–a twS = ∞•~Io	190 1/(c–a) wS = t(∞•~Io)	100 b/(c–a) wF intersection with circumcircle	101 b2/(c–a)
519 c+a–2b ∞•(G—Io)	903 1/(c+a–2b) t(∞•(G—Io))	88 b/(c+a–2b)	106 b2/(c+a–2b)
513 b(c–a) ∞•~tIo	668 1/b(c–a) t(∞•~tIo)	190 1/(c–a) wS	100 b/(c–a) wF = focus of Yff parabola
900 (c–a)(c+a–2b) (∞•~190o)	t900 1/(c–a)(c+a–2b) t(∞•~190o)	? b/(c–a)(c+a–2b)	901 b2/(c–a)(c+a–2b)
812 (c–a)(b2–ca)	? 1/(c–a)(b2–ca)	660 b/(c–a)(b2–ca)	813 b2/(c–a)(b2–ca)
? (c+a)(b2–ca) ∞•(Io—tIo)	? 1/(c+a)(b2–ca) t(∞•(Io—tIo))	g2238 b/(c+a)(b2–ca)	? b2/(c+a)(b2–ca) g(∞•(Io—tIo))
(a2–bc+c2–ab)(c–a)
? b(c2+a2–ab–bc)	? 1/b(c2+a2–ab–bc)	673 1/(c2+a2–ab–bc) wT	105 b/(c2+a2–ab–bc)
? b(c–a)sbb,	? 1/b(c–a)sbb,	658 1/(c–a)sbb,	? b/(c–a)sbb,
	From the Gergonne point also the Mittenpunkt
g109 (c–a)sb ∞•~Go	r109 1/(c–a)sb	651 b/(c–a)sb	109 b2/(c–a)sb
? csc+asa–2bsb ∞•(G—Go)	? 1/(csc+asa–2bsb) t(∞•(G—Go))	? b/(csc+asa–2bsb)	? b2/(csc+asa–2bsb)
	From the Symmedian point
523 c2–a2 tS = ∞•~K	99 1/(c2–a2) t(∞•~K) the Steiner point	662 b/(c2–a2)	110 b2/(c2–a2) the focus of the Kiepert Parabola
524 c2+a2–2b2 ∞•(G—K)	671 1/(c2+a2–2b2) t(∞•(G—K))	897 b/(c2+a2–2b2)	111 b2/(c2+a2–2b2) isogonal of G—K endpoint
512 b2(c2–a2) gS = ∞•~tK	670 1/b2(c2–a2) rS = t(∞•~tK)	799 1/b(c2–a2) t of Io–Mineur conic perspector	99 1/(c2–a2) Steiner point
g98 b2SB2– gT	r98 1/b2SB2– rT	1821 1/bSB2–	98 1/SB2– Tarry point T
	From the Orthocenter also the Circumcenter
? (c2–a2)SB ∞•~H = ∞•~O	? 1/(c2–a2)SB	162 b/(c2–a2)SB	? b2/(c2–a2)SB
30 SBC+SAB–2SCA ∞•(G—H) infinite point on Euler line	1494 1/(SBC+SAB–2SCA) isotomic of Euler endpoint	b/(SBC+SAB–2SCA)	74 b2/(SBC+SAB–2SCA) isotomic of Euler endpoint
? b2(c2–a2)SBB	? 1/b2(c2–a2)SBB	823 1/b(c2–a2)SBB	? 1/(c2–a2)SBB
	Fissile points
	r1113, r1114 b2(J–1)SB+2bSCA D–tO meets Steiner ellipse	2580,2581 b (J–1) SB+2SCA / b Wo—tWo meets Io circumonic	1113,1114 b (J–1) SB + 2 SCA Euler meets circumcircle
	not so sure what to make of these
? b(c–a)sb(4SBB–c2a2)	? 1/b(c–a)sb(4SBB–c2a2)	655 1/(c–a)sb(4SBB–c2a2)	b/(c–a)sb(4SBB–c2a2)
		1492 b/(c–a)(c2+a2+ac)

Locus property

If a point is on an I_x-circumconic, the Newton line of its Cevian quadrilateral is tangent to the circumcircle.

All four circumconics

These conics are generated as the isotomic conjugates of ~I_x, the duals of the four incenters. These four lines intersect 6 times, two on each triangle edge, a given line meeting each of the others at a different edge. These conics only intersect at vertices.The fourth intersections of the conics are double intersections at a vertex. Each conic is tangent to the the 3 others, each at a different vertex.

Since each conic is tangent to an internal or external bisector at a vertex, and since at a vertex these bisectors are perpendicular, these conics meet orthogonally or parallelly at the vertices, each conic being orthogonal to two and parallel to one.

The endpoints of the lines map onto the intersections of these conics with the Steiner ellipse.

Figure: Each version of the 4 I_x-circumconics is given a different color. The 4 generating lines, the duals of the I_x, are shown dashed. The points are shown in all versions of these conics. The quartile points are shown, each on its version of the conic.

Perspectors and Centers

The perspectors are the extraversions of I_o = (:b:) which are harmonic associates as shown in the picture below. The harmonic associates are one each in each region of the plane; hence only one perspector can be inside the triangle and inside the Steiner inellipse. Hence it is only possible for one conic to be an ellipse.

The lines I_x—M_x concur at K.

The ~I_o lines meet at the traces of the isotomic conjugates of the perspectors.

The centers are the Mittenpunkts, obtained by operations mtd of the perspectors, an harmonic set of 4, which guarantees that they will be desmic . Here M_abc = : b/s_b : is the desmic mate of M_o = : b s_b : .