Eulerian and Brocard

The Euler lines of I_oBC, AI_oC, ABI_o, where I_o is the original incenter, concur at the Schiffler point H_o, on the Euler line of ABC. Successively replacing I_o by other members of its family I_a, I_b, and I_c generates the extraSchiffler points H_a, H_b, H_c. All told there are 12 lines which meet three at a time at 4 Schiffler points on the Euler line of ABC. This is an example of the principle that if one member of a weak set of points lie on a strong line, such as the Euler line, then they all must be on that line. This essay is about some weak points on strong lines and the circumcircle and the relationships thus developed.

The Schiffler point discovery is one of those wonderful episodes in triangle geometry that has grown way past the original discovery. It was rapidly found that if the incenter was replaced by other points, including the orthocenter and circumcenter, the Euler lines formed again concur on the Euler line of ABC, eventually leading to the discovery of all the points the produce triangles whose Euler lines concur. These were the points on the Neuberg cubic, the circumcircle, and the line at infinity. Then it was discovered (by me I think, and proved by Lamoen and Conway) that the four Brocard lines concur as well, for points on the same cubic.

But once a cubic is found, I get interested in lines on the principle that the job of the cubic is to organize those lines, so they must be interesting. And they are!

There are other interesting intersections of these 12 lines. We will investigate those, eventually being led to projective and affine properties of the incenter itself.

An analogous structure exists for the 12 Brocard lines of the same triangles which concur at the isogonal Spieker points, all on the Brocard line of ABC. (Note: if isogonal weak points lie on a strong structure (the OK line in this case), so do the isotomic ones, since isogonal -> isotomic is a projective transformation, and is strength preserving. In this case the isotomic Spiekers lie on the G—K line, the symmedian track).

The following statements summarize the two situations:

The 12 Euler lines meet 3 at a time at the original Schiffler point H_o ~ : b s_b/(c+a): and its 3 extraversions, (known as the Schiffler points) all of which are on the Euler line of ABC.

The 12 Euler lines also meet 3 at a time at Q_o ~ : b/(c-a) : [X(81)] and its 3 extraversions, all of which are on the circumcircle.

The 12 Euler lines meet 2 at a time at the 6 midpoints of the lines connecting I_o, I_a, I_b, I_c, all of which are on the circumcircle.

The 12 Brocard lines meet 3 at a time at the isogonal-Spieker point gS_o ~ : b²/(c+a) : [X58] and its 3 extraversions, (known as the co-Spieker points) all of which are on the Brocard line of ABC.

The 12 Brocard lines also meet 3 at a time at Q'_o ~ : b²/(c-a) : [X(101)] and its 3 extraversions, all of which are on the circumcircle.

The 12 Brocard lines meet 2 at a time at the 6 midpoints of the lines connecting I_o, I_a, I_b, I_c, all of which are on the circumcircle.

I like parallel structure and I like projective configurations. The similar nature of the formulas is also intriguing: factors of c+a inside and c–a ourside the triangle.

There are more properties of these points. Properties communicated to me by Darij Grinberg are indicated with (DG).

1. The traces of lines oa, ob, oc on their respective sides are perspective to ABC, with perspector tS_o = :1/(c+a): , the isotomic Spieker point on G—K. There are 4 such points (DG).
2. The trace of oa is also on I_a--G. There are 3 such lines (DG).
3. tS_o lies on the line joining the incenter and the isotomic incenter. There are 4 such lines (DG)
4. oa, ob, and oc concur at H_o, the original Schiffler point, on e, the Euler line. There are 4 such points.
5. aa, bb, cc concur at : b/(c-a) : on the circumcircle. There are 4 such points.
6. ob and bb meet at I_o + I_b, the midpoint of I_o and I_b, also on the circumcircle. There are 6 such points.
7. The retros of the points in 5 lie on the Steiner ellipse, the lines connecting corresponding points of 5 and 7 concur at the Steiner point S (which of course must be true since all these lines concur at S. [Note: the "retro" operation takes is the projctive operation that fixes ABC and takes the circumcircle to the Steiner ellipse.]

This table gives properties of the weak points mentioned on this page, all of which are on Strong lines or strong conics. The designation "oa" refers to the Euler (or Brocrd) line of using the original incenter, subtriangle of the "a" type.

name	notation	coordinates	line	comments
Brocard concurrence point	gS_{o X(58)}		oa, ob, oc	lies on OK; lies on the line I_o—H_o
Euler concurrance point	H_{o X(21)}		oa, ob, oc	Schiffler's original discovery. on Euler line on I_o–gS_o line on S_o–X81 line
Brocard correspondenc point	Q'_{o X(101)}		aa,bb,cc	isogonal conjugate of ~I_o•∞; on circumcircle;
Euler correspondence point	Q_{o X(100)}		aa,bb,cc	isogonal conjugate of ~tI_o•∞; on circumcircle; on I_o circumconic; dF_o
Io—Ib midpoint	I_o + I_b		ac, ca	both individual Euler and Brocards meet here; on circumcircle
perspector	_X(81)	: b/(c+a) :	on GK	on I_o—H_o—gS_o
perspector	tS_{o X(253)}	: 1/(c+a) :	on GK	on I_o—tI_o on X190-X37 line
Steiner projection	rQ_{o X(190)}	: 1/(c–a) :		isotomic conjugate of ~I_o•∞; on Steiner ellipse; on Q_o—S
Steiner projection	rQ'_o	: 1/b(c–a) :		isotomic conjugate of ~tI_o•∞; on Seiner ellipse; on Q'_o—S
	X(37)	: b(c+a) :		on X190 - tS_o

I like this grouping for several reason: the nice correspondence between the Brocard and Euler situations and for the different ways that these weak points all lie on strong structures yet still manage to be interconnected. Some of these points come from the infinite points on the dual of I_o. The next section explores this connection.

The 12 Euler lines also have a property not matched by the Brocard lines. They meet 2 at a time at 12 points on the edges of ABC (four on each edge), each line going between two such points. I do not know what to make of this last property.

Properties of G—I_o and the dual line of I_o

The dual line (the tripolar of the isotomic conjugate) to I_o has line coordinates [:b:] which meets the point at infinity at (: c–a :). The isotomic conjugate of this point is (:1/(c-a):) is on the Steiner ellipse. The isogonal conjugate of this point is (: b²/(c-a) :), which is on the circumcircle and is called the "Brocard correspondence point" above. These last two points are colinear with S.

The Ex/Extraversions of I_o lead to a new point. The b-trace of G—I_b on the edges is ( 1/(b+c) : 0 : 1/(a+b) ) so that the corresponding points have perspector ( :1/(c+a): ). The trace of the Euler line ob goes through the same point and leads to the same perspector. This begins a pattern of correspondences between points with factors of c–a, which are on circumconics, and points with factors of c+a, which are on strong lines in the triangle interior.

Each point at infinity has a corresponding point on the Steiner ellipse (its isotomic conjugate) and one on the circumcircle (its isogonal conjugate). These two points are always colinear with the Steiner point, which is interesting. The operation that does this is projective, the "pro" operation : y : -> : b²y :. This operation also takes the symmedian track G—K into K—P^o (= pK) which is the Brocard meridian line O—K.

Other conjugates give other conics. The point (l:m:n) determines a conugation,which takes a point from the line at infinity to a circumconic, then the projective operation :y: -> : my : takes a point from the Steiner ellipse to the corresponding point on this conic. The lines between the two points concur at : 1/(n-l) : on the Steiner elllipse, which is the fourth intersection of the conic and the Steiner ellipse.

For our circumstance many of the points are generated by the incenter so the Incentral circumellipse (center = M_o, the Mittenpunkt), the Steiner ellipse, and the circumcircle are the conics of interest to us.

The central points related to the incenter follow a pattern derived from the orbit of I_o in a group defined by Clark Kimberling 10 years ago. This orbit begins and ends on a vertex and goes through the centroid. Many points are in a band around it that I call the "sweep" of the incenter. A dramatic version of the "sweep of the incenter" is shown in the next picture. It shows the pattern formed by points of the barycentric form : b_ⁿ(c+a)_^m : . They lie around a very well defined path. Of course this includes many of the points we are discussing. A much wider variety of points shows the same pattern which I call "the island of the incenter." Many points lie around this basic path. This tendency of of triangle points to follow (approximately) this path is also shown in later pictures with the points we are studying.

Figure: The c+a sweep. This illustration uses points with coordinates of the form b_ⁿ(c+a)_^m_^..

The next picture shows many of the points listed in the chart above. The sweep of the incenter is shown as a light blue path through the triangle. In addition the extraversions of the Shiffler points and the co-Spieker points are shown.

This figure shows a lot of things. Along the red Steiner circumelllipse are the conjugates of the endpoints of the natural lines generated by Io: ~Io, the dual of Io, and G—Io. These points are opposite on the Steiner ellipse. Analogous points are shown for tIo. The sweep of the inceter is indicated in light blue in the interior of ABC.

Here is a summary of properties, some contributed by Darij Grinberg. I will abbreviate the Euler line, the Brocard line, and circumcircle of ABC by e, b, and the circumcircle. The individual Euler and Brocard lines will be given by a two letter designation such as xy, where x = o,a,b,c indicates which incenter is uses and y = a,b,c indicates which triangle. X—Y will denote the line between points X, Y.

Now what do we make of all this. There is a group of point in the central region with factors of (c+a) in their coordinates. There are points related to the three circumconics that have factors of (c-a) in their coordinates. Well there are connections between these groups.

For example : c–a : which is the infinity point on and : c+a :, the Spieker point, go through are harmonically conjugate with :c: and :a: which are rotated versions of Io which I notate as Io> and Io<. This means the the Spieker point is the midpoint between these two and the line between them is parallel to the dual of I_o.

The isotomic Spieker point :1/(c+a) : ~ : bb + (bc + ca + ab) : is on GK. These coordinates can be rearranged as ~ : 2 b so + ca : so that this point is also on the Io—tIo line.

Similarly :1/(c-a) : ~ : -bb + (bc – ca + ab) : is on K—parallelians, and on Mo—tIo.

The points X37 ~ b(c+a) and X894 ~ : bb + ca : are harmonically conjugate to the last two points.

Similarly :b²/(c+a) : ~ : b⁴ + b²(bc + ca + ab) : is on the Brocard Meridian Line, and on X41—I_o.

Similarly :b²/(c–a) : ~ : –b⁴ + b²(bc – ca + ab) : is on pK—pro-parallelians, and on pN_o—I_o.
pN_o is the "pro-Nagel" point, which is one of the centers of similitude of the incircle and circumcirce and, hence, is on the I_o—O line.

The points b³(c+a) and : b²(b² + ca) : are harmonically conjugate to the last two points but neither seem to be in ETC.

Similarly :b/(c+a) : ~ : b³ + b(bc + ca + ab) : is on the I_o—pI_o line, and on G—K.

Similarly :b/(c-a) : ~ : –b² + b(bc – ca + ab) : is on X31—:b(bc – ca + ab): (not in ETC) and on M_o—G.

The points b²(c+a) = pS_o (= X42) and : b(b² + ca) : (not in ETC) are harmonically conjugate to the last two points.

Figure: The paths through and around ABC for points having (c+a) in their coordinates, which are in the central region, and (c–a) in their coordinates, which move between 3 circumconics.

Positions of the Schiffler points

The b barycentric coordinate for the Schiffler point is

b s_b / (c+a) ~ (a+b)(b+c) b s_b ~ b²(c² + a² – b²) + (a + b + c) abc

This expression is nicely written as vectors, where O and G are the vectors representing those points. Here I use Conway notation where s_o = (a+b+c)/2, S = 2 area.

2 S² O + 3 abc s_o G ~ 2 O + 3 R/r_o G which nicely gives the known ratio of distances GH_o/ H_oO = 2r_o/3R. Note: this method converts the homogeneous coordinates (which describe projective properties only) to vectors with coefficients, which describes relative postion, an affine property. For centrally define points, the coefficients are symmetric functions in triangle variables, such as r_o and R.

We know that R/r_o varies from 2 (for equilateral triangles) to infinity (for highly obtuse ones), which means that the Schiffler point is 1/4 of the way from G to O for nearly equilateral triangles and very close to G for obtuse ones. This means that it does not wander much on the Euler line and sticks near G.

The other Schiffler points are found as follows. If we extravert in the B manner R -> R, r_o -> – r_b, so that the positions of the B-Schiffler looks like 2 O – 3 R/r_b G. All three extraSchifflers are outside of the interval GO, but on which side?

If the triangle is equilateral then r_b/R is 3/2. As we move vertex B so that B becomes the largest angle, this ratio increases for r_b/R but decreases for the other two circles.

Hence in the extraversion of the formula quoted above GH_b/ H_bO = –2r_b/3R, we notice that for nearly equilateral triangles GH_b/ H_bO = –1, so that the extra-Schiffler points are all at the harmonic conjugate of the middle of GO, ie, at infinity. As angle B increases, this ratio increases for r_b, but decreases for the other two. This means that H_b is on the O side of GO, while H_a and H_c are on the G side of GO. More generally the extraSchiffler point corresponding to the largest angle is on the O side of GO.

We can give an idea of how far away H_b is for large B angles. Since r_b/R <= 4. GH_b/H_bO < –8/3.

This was fun

[Darij] Subject: An affine generalization of the Schiffler point

It is well-known that if I is the incenter and Ia, Ib, Ic are the excenters of a triangle ABC, then
(1) the Euler lines of triangles BIC, CIA, AIB concur at the Schiffler point X(21).
(2) the Euler lines of triangles BIaC, CIbA, AIcB concur at the point X(100).

These both properties can be generalized as follows:

Let P be a point with homogeneous barycentrics ( x : y : z ), and let PaPbPc be its anticevian triangle. Call Ma, Mb, Mc the midpoints of PPa, PPb, PPc, call Ga, Gb, Gc the centroids of the triangles BPC, CPA, APB, and Ha, Hb, Hc the centroids of the triangles BPaC, CPbA, APcB.

Then,

(1) the lines MaGa, MbGb, McGc concur at the point with homogeneous barycentrics

/ x(y+z-x) y(z+x-y) z(x+y-z) \
( -------- : -------- : -------- ).
\ y+z z+x x+y /

(2) the lines MaHa, MbHb, McHc concur at the point with homogeneous barycentrics
/ x y z \
( --- : --- : --- ).
\ y-z z-x x-y /

These affine theorems indeed generalize the Schiffler results, because:

In the case P = I, we can easily show that Ma, Mb, Mc are the circumcenters of triangles BIC, CIA, AIB, but also the circumcenters of triangles BIaC, CIbA, AIcB simultaneously. Then we infer that the lines MaGa, MbGb, McGc are the Euler lines of triangles BIC, CIA, AIB, while the lines MaHa, MbHb, McHc are the Euler lines of triangles BIaC, CIbA, AIcB.

By the way, for every P, if Na, Nb, Nc are the midpoints of segments BC, CA, AB, then the lines MaNa, MbNb, McNc also concur, and the point of concurrence has homogeneous barycentrics
( x(y+z–x) : y(z+x–y) : z(x+y–z) ).

For P = K, this is the circumcenter of triangle ABC.
Darij Grinberg

Eulerian and Brocardian concurrances

Schifflers and coSpiekers


	Eulerian and Brocardian concurrances Schifflers and coSpiekers