The Excentral Triangle

Introduction

For any triangle ABC we know that there are two perpendicular angle bisectors at each vertex. These 6 bisectors meet 4 times, giving 4 triangles. The most central of these is the excentral triangle I_aI_bI_c. Altitudes of the excentral triangle are the angle bisectors of ABC, making ABC is the orthic triangle of ∆^Io = ∆I_aI_bI_c. Since the bisectors at a vertex are perpendicular, quadrilaterals such as I_oAI_bC are cyclic (see figure). The centers of these six circles are all on the circumcircle of ABC. This allows us to easily fill in the angles for the excentral triangle.

The excentral triangle has vertices I_a, I_b, I_c. Its altitudes are the internal angle bisectors of ABC so that H of ∆^Io is I_o of ABC. This is the first indication that strong points of  ∆^Io are weak points of ABC.

The notation here was invented by John Conway. Points referred to a Cevian triangle get the perspector added as a subscript. Points referred to an exCevian triangle get the perspector as a suffix. Hence H^Io is the orthcenter of the excentral triangle.

Since so much of the discussion of the strong centers of the excentral triangle will be a discussion of the weak points of ABC, Figure 2 is a picture of the weak points, such as the Spieker, Gergonne, Nagel, and Mittenpunkt points of ABC, embedded in the excentral triangle. Here is a link to Triangle Book notation, which is used here.

The weak points M_o, N_o, S_o, and M_o (colored light red) are connected by lines each of which join to a strong point (colored blue) . The significance of the strong point is that all extraversions of that line will go through the same point. The Clawson cotangent point W_o is on two of the lines. W_o and I_o are colored differently because, although weak points, they are very special ones as their harmonic associates (the "ex" version) are the same as their extraversions. The lines I_x—G_x (x = o,a,b,c) concur at L, the deLongchamps points, which is out of the picture below. Perpendiculars through I_x to the edges of ABC, concur at oI_o, the reflection of the incenter in the circumcenter.

For our purposes the most significant line will be the W_o—M_o—G—G_o—gM_o, where g indicates the isogonal conjugate operation, as well as the line  oI_o—O—I_o—gM_o.

These properties of the weak points of ABC will explain most of the properties of the strong points of the excentral triangle, which become the weak points in ABC. They form lines whose properties are governed by the structure relating weak and strong points in this picture.

edgelengths of excentral triangle

There is a really nice way to find the edgelengths of the excentral triangle. We know that ABC is the orthic triangle for the excentral triangle and that if a, b, c are the sides of a triangle then the sides of the orthic triangle are proportional to a² S_A, b² S_B, c² S_C. Let x, y, z be squares of the sides of the exceentral triangle triangle so that up to proportionalities, we have x(-x+y+z) = a, y(x-y+z) = b, and z(x+y-z) = c which solves easily to give
x = a s_a,  y = b s_b, and z = c s_c, which gives the formulas we give below.

Ia—Ib edgelength

The edgelengths of the excental triangle look like

= =λ where λ = .

and similarly for the other edges. The parameter lambda is a parmeter dependent on the shape of ABC; it varies from ∞ (degenerate triangles) to 2 √2 (equilateral).

The product of the three edgelengths = abc

Io—Ib edgelength

Its perpendicular distance I_o—I_b is the b-extraversion of the above

Since BI_o = r_o csc B/2 = , we have that the area of I_oI_aI_c = 4R s_b, so the the area of the excentral triangle is 4Rs_o (although I keep worrying about factors of 2).

important functions of the edgelengths

which means that the squares erected on the sides are divided into sections proportional to

which is the same as 2 abc a, which means that the sides of the extangent's orthic triangle are proportional to the edges of ABC. This is obvious because the orthic triangle of the extriangle is ABC, and we know that the a-side of the orthic triangle is proportional to (+-)where a,b,c are the edgelength. This is a reason that many centers of this triangle have simple coordinates.

which is proportional to .

Centers of the excentral triangle

The theme here is that the strong points of the excentral triangle become commonly recognized weak centers of ABC.

If you look at the ETC coordinates for points related to the excentral triangle (extriangle for short), most depend on a trig function half angles such as tan B/2. tan B/2   ~  s_ca, so that points with this type of dependence relate well to the others of the X-points, and Kimberling gives many relationships for these points. For X(165), the centroid of the excentral triangle (trilinears : tan(B/2) + tan(C/2) - tan(A/2)); I get barycentric coordinates  ~   : b(s_ca - b s_b)  : . This form immediately tells us that this point is related to the co-mittenpunkt and the direct center of similitude of the in and circumcircles.

But the coordinates of the incenter of the extriangle use its edgelengths, which are proportional to   . These coordinates make it difficult for this point to "talk" to points that do not have this radical in it (which, since b is itself a radical, is a square root of a square root). For these points their algebraic nature is a determining factor in their analysis.

This means that strong points of the extriangle, whose coordinates depend on squares of the edgelengths of that triangle, relate nicely and are friendly to the greater community of triangle points. But weak points of the excentral triangle will be algebraically prevented from these same connections, and are largely condemned to live in a world with few connections.

I compute that the incenter of the extriangle has the form.

Except for a few points with similar coordinates, this point is only going to simply relate to a small number of others. See the supplementaire page for some examples of related points.

Excentral Centers to ABC Centers

From Peter Moses;
Just to continue a wee bit, ETC says X(504) lies on line X(164) X(173). Also on this line is X(1128). This happens to be X(1844) of the Excentral triangle. I also note that line X(1) X(19) of the excentral triangle is line X(164) X(173) of ABC. X(503) is X(92) of the excentral triangle.   I don't know how interesting this list is, but they are X(n) for the Excentral triangle = X(m) ABC.  

[Note: I have added to Peter's list by emphasizing the points who have square roots of square roots and are my candidates for sparsely connecting points. I did this by searching through Kimberling's list for appropriate functions of the angle A/2]

For example the first entry, {1,164} means, X(1) Excentral = X(164) ABC. It presumably also means that X(164) Orthic = X(1) ABC.
{1, 164}, {2, 165}, {3, 40}, {4, 1}, {5, 3}, {6, 9}, {7, 166}, {8, 167}, {9, 168}, {15, 1277}, {16, 1276}, {19, 173}, {24, 46}, {25, 57}, {31, 362}, {32, 169}, {33, 258}, {46, 505}, {48, 504}, {51, 2}, {52, 4}, {53, 6}, {54, 191}, {57, 363}, {63, 845}, {64, 2136}, {65, 188}, {68, 1490}, {69, 2951}, {75, 844}, {76, 170}, {92, 503}, {95, 2938}, {96, 2939}, {97, 2941}, {98, 1282}, {107, 1054}, {110, 1768}, {113, 104}, {114, 103}, {115, 101}, {122, 1293}, {125, 100}, {127, 1292}, {128, 74}, {129, 98}, {130, 99}, {131, 102}, {132, 105}, {133, 106}, {134, 107}, {135, 108}, {136, 109}, {137, 110}, {138, 111}, {139, 112}, {143, 5}, {155, 84}, {158, 361}, {184, 63}, {185, 8}, {186, 484}, {216, 573}, {225, 266}, {230, 910}, {231, 2173}, {232, 672}, {235, 56}, {249, 2958}, {250, 2957}, {252, 2940}, {254, 2956}, {264, 1742}, {275, 846}, {317, 1721}, {389, 10}, {393, 1743}, {403, 36}, {418, 1764}, {421, 1758}, {427, 55}, {428, 354}, {429, 260}, {460, 241}, {467, 1754}, {468, 1155}, {546, 1385}, {570, 71}, {571, 19}, {577, 1766}, {647, 649}, {847, 1745}, {973, 442}, {974, 1145}, {1093, 978}, {1105, 2943}, {1112, 11}, {1113, 2448}, {1114, 2449}, {1136, 1507}, {1141, 2948}, {1147, 1158}, {1263, 399}, {1312, 1381}, {1313, 1382}, {1560, 2291}, {1562, 644}, {1593, 1697}, {1594, 35}, {1596, 999}, {1609, 2270}, {1637, 1635}, {1799, 2942}, {1824, 174}, {1826, 259}, {1827, 1488}, {1828, 2089}, {1829, 177}, {1838, 1130}, {1843, 7}, {1844, 1128}, {1856, 289}, {1879, 48}, {1899, 200}, {1968, 2082}, {1974, 1445}, {1986, 80}, {1990, 44}, {1993, 1709}, {2052, 43}, {2072, 2077}, {2165, 610}, {2262, 236}, {2351, 1763}, {2489, 657}, {2501, 650}, {2679, 927}, {2904, 90}, {3003, 2183}, {3018, 2265}      

More about this list soon.

Soon is now. The first thing to do is to pick out the points in the excentral triangle that become well known weak points in ABC. Here are some examples: {2, 165}, {3, 40}, {4, 1}, {6, 9}, {1843, 7}, {185, 8},{389, 10}. The relavance there is that these points will probably pair to form four lines concurring at a strong point of ABC. For example O^Io and H^Io of the excenral triangle will have four versions whose lines, the Euler lines of the excentral triangle, which concur at O of ABC (see description and pictures later). The OK lines of the excentral triangle will come in four versions and concur at H. The 185-1843 (whatever this line is, will concur at D). X389 is the Taylor center which is known to be the Spieker center of the orthic triangle.

Another thing to check is the nature of the points that become strong in ABC such as: {51, 2}, {52, 4}, {53, 6}, whose trilinears depend on double angles and barycentric depend on fourth degree or higher strong polynomials. In these cases they are listed as strong points of the orthic triangle. This makes sense since ABC is the orthic triangle of ∆^Io .

The Euler lines

Many of the strong points we will show, the ones whose coordinates are most reasonable, will be points on the Euler line of the excentral triangle. This means that there will be 4 little-Euler lines, all through O, it turns out.

Centroids

The centroid of the excentral triangle has a large number of related points. There are the harmonic associates of the centroid, as well as the extraverted triangles such as I_a, I_b, I_c which have harmonic associates of their own. All told there should be a total of 16 such points.

The centroid of the excentral triangle is computed from the normalized sum of I_a, I_b, I_c. It has a very nice form which relates it to the isogonal conjugate of the Mittenpunkt, which is the desmic mate of the Mittenpunkt, and the direct center of similarity of the circumcircle and the incircle.
The point    : b + : is obviously of interest and related to these points.

The b-excentroid of the excentral triangle is the normalized sum of I_a, -I_b, I_c. It is  

which, by its structure, can be seen to form a desmic system with ABC.
The centroid of the b-extraverted triangle I_a, I_o, I_c is

incenters

I hope to have more to say about this point eventually. Its form is a bit confusing since its extraversions will be imaginary (actually they will not since I divided out a factor of √abc, which if restored solves the extraversion problems. Since so much of the discussion of the strong centers of the excentral triangle will be a discussion of the weak points of ABC, here is a picture of those weak points of ABC embedded in the excentral triangle. Here is a link to Triangle Book notation, which is used here. .

This point has the form of the "complementaire" and must be colinear with points with this form. Kimberling lists some of them and these points are colinear, but these other points have nothing to do with the incentral or excentral triangles. The angle form of the above is, in trilinears,  sin B/2 + sin C/2 - sin A/2, a supplementaire. These points are colinear X266  ,  X1, -- X164              258 is on this line.

circumcenters

circumcenter of excentral triangle  

which makes it obvious that this is the reflection of the incenter in the circumcenter of ABC. This point forms a desmic system with ABCL and its isogonal conjugates.

symmedians

The symmedians form a double desmic system around the 16 symmedian points and their extraversions. I believe that this is typical of the strong points of the excentral triangle that form weak points of ABC. This situation is the same for them all. There are 4 versions of the excentral triangle, the original IaIbIc and ones like IaIoIc. Each of these triangles has 4 symmedian and ex-symmedian points, making 16.

I had realized that a double-desmic system was possible a number of year ago, but never expected to see it arise in an interesting context. It is a maximally interconnected system of 24 points that arises inevitably from a desmic system that uses 16 points (the 12 of the system and 4 Harmon points). I found that each desmic system, which is held together by a single strong point (the desmon) and 4 weak points of a special type (the Harmons) could be reversed into a new desmic system with the roles of the Harmon and Desmon are reversed. This circumstance occurs in precisely our situation here.

Hopefully I will be able to create a picture of this complicated system.

The first desmic system

The symmedian point of the excentral triangle is the Mittenpunkt of ABC.

The symmedian of Ic, Io, Ia is

which is the extraversion of the Mittenpunkt and is known to form a desmic stystem with the co-Mittens, their isogonal conjugates, and ABCG. In this case the Harmon point is the Clawson point : b SB :  The three points of this form are perspective to ABC with perspector gM_o, which is shown by arrows in the picture below.

The second Desmic system.

The b-exsymmedian point is the b-extraversion of the Mittenpunkt

which gives nice relationships between the I's, G, the clawson pts and these points

The b-exsymmedian of Ic, Io, Ic is the b-exMittenpunkt

These two sets of points form a desmic system, but this time the Clawson point is the desmon and G and its harmonic associates the harmon points.

Together with the first set, there are 24 interconnected points in this system.

Comments

Here is a picture of many of the points listed in the last section.

In this picture the lines oI_o—G^Io—I_o and its extraversions are the Euler lines (O—G—H) of the excentral triangle and its extraversion. All four concur at O, the circumcenter of ABC and go through one of the isogonal Mitten points (gM_x for x = o,a,b,c).

The arrows show that the three of the excentral symmedian points concur at gM_o. This is the concurrance needed to cause a desmic system involving the 4 symmedian points. But see the above note about the symmedian points and the multiple extraversions.

Note that the incenters in the picture to do not relate to any of the other points.

--------------------

More about X(504) and X(505) and the incenter of the excentral triangle. These points began my investigations of the incentral and excentral triangles.

These points have coordinates that depend on quantities that are roots of roots, such as   root(b s_b).

So if one is to take these points seriously, one would look for points that have these factors in their coordinates. One would do for these as Kimberling has done for normal points: start generating and numbering points with coordinates of this form. One would get another set of geometry properties, with a bit uglier algebra, but it would be a different one because the field extensions (extraversions) would generate a more complicated structure (some of it imaginary) that should be interesting.

This world would have new parameters, analogous to the Brocard parameter that controls so much of the triangle geometry that we are used to.

The positions of centers on lines is controlled by symmetric functions in the sides of the triangle. Except for s, all the symmetric functions simply tell us how the Brocard angle positions the points in relation to each other. We get the Brocard angle from  ++= 2S cot Ω, and all higher degree symmetric functions depend on it.

If the coordinates of a point depend on, say, root(b sb), then symmetric functions of this quantity will control much of the geometry and will have less to do with the Brocard angle.

Created by Mathematica  (September 25, 2005)