brisse 3rd.nb

Third degree points

This section gives symmetric versions of the 3rd degree points from Edward Brisse's collection taken from Kimberlings collection. This symmetrrization often dramatically simplifies the coordinates and takes them from meaningless polynomials in a, b, c to more meaningfull ones in symmetric functions of a, b, c. Since the algebra of coordinates mirrors the geometry of points, and since considerations of symmetry are important, the algebra of geometry should reflect the same desire for symmetry.

Not all points simplify, so they are not given. Some simplify so badly that I figured that this might be important, and I listed them.

There are many ways to symmetrize 3rd degree polynomials so I tested a number of different possibilities. Sometimes several possibilities are given.

Some things I have noticed: symmetric parts are usually sums of abc and , and never, say, s_o³. The Clawson cotangent point, and the center of similarity of the circumcicle and incircle points show up frequently. Many 3rd degree points are on the lines from G to these points. I have known about the first of these lines for a long time, but the second was a surprise.

In The Triangle Book, we call X(67, if I remember correctly) the Clawson point. He originally found the functions that are named after various trig functions. The name Clawson was appended to one, the not so important one, we think, so we have given his name to the one that shows up the most in what we do.

Note: I have not yet checked what points these are that are on these lines. They might be some of the silly points in the compilation. But since the complicated coordinates simplify so well, probably not.

Points on the G—direct center of similarity line

X11, X149, X398, X497, X528, X918, X1376, X1621, X2550

Points on the G—Clawson cotangent point line

X329, X553, X614, X908, X2094

X11

This point is one with simple coordinates, but it is still useful to have a representation in products and one in sums.

check

X42

This one is here to show that some points are best left as is. This also shows one of my choices in symmetric functions to use.

-1/2 a b c + a^2 s_a + s_abc + (a b + a c - b c) s_o + s_a s_o^2 - (3 a S_A)/2

-1/2 a b c + a^2 s_a - s_a s_o^2 + s_o^3 - (a S_A)/2

check

X149

A nice simplification. This point is on the G—COS line.

asymmetrize[a^3 - a^2 * b + a * b^2 - b^3 - a^2 * c - a * b * c + b^2 * c + a * c^2 + b * c^2 - c^3//Expand]

X165

asymmetrize[a * (3 * a^2 - 2 * a * b - b^2 - 2 * a * c + 2 * b * c - c^2)//Expand]

X329

Again a nice simplification. This point is on the G—Clawson line.

asymmetrize[a^3 + a^2 * b - a * b^2 - b^3 + a^2 * c - 2 * a * b * c + b^2 * c - a * c^2 + b * c^2 - c^3//Expand]

a^3 + a^2 b - a b^2 - b^3 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2 - c^3

X354

asymmetrize2[a * (a * b - b^2 + a * c + 2 * b * c - c^2)//Expand]

X398

asymmetrize[(a - b - c) * (3 * a^2 + b^2 - 2 * b * c + c^2)//Expand]

X497

asymmetrize[(a - b - c) * (a^2 + b^2 - 2 * b * c + c^2)//Expand]

X516

asymmetrize[2 * a^3 - a^2 * b - b^3 - a^2 * c + b^2 * c + b * c^2 - c^3//Expand]

X518

asymmetrize[a * (a * b - b^2 + a * c - c^2)//Expand]

X528

asymmetrize[2 * a^3 - 2 * a^2 * b + a * b^2 - b^3 - 2 * a^2 * c + b^2 * c + a * c^2 + b * c^2 - c^3//Expand]

X537

asymmetrize[a^2 * b - 2 * a * b^2 + a^2 * c + b^2 * c - 2 * a * c^2 + b * c^2//Expand]

(a b c)/2 + 3 a^2 s_a + s_abc - (a b + a c - b c) s_o + s_a s_o^2 - (7 a S_A)/2

X553

asymmetrize[(a + b - c) * (a - b + c) * (2 * a + b + c)//Expand]

X612

asymmetrize[a * (a^2 + b^2 + 2 * b * c + c^2)//Expand]

a b c - 2 a^2 s_a + 2 s_abc + 2 (a b + a c - b c) s_o + 2 s_a s_o^2 - a S_A

asymmetrize2[a * (a^2 + b^2 + 2 * b * c + c^2)//Expand]

X614

asymmetrize2[a * (a^2 + b^2 - 2 * b * c + c^2)//Expand]

X846

asymmetrize[a * (a^2 - a * b - b^2 - a * c - b * c - c^2)//Expand]

-1/2 a b c - a^2 s_a - s_abc - (a b + a c - b c) s_o - s_a s_o^2 - (a S_A)/2

asymmetrize2[a * (a^2 - a * b - b^2 - a * c - b * c - c^2)//Expand]

-1/2 a b c - a^2 s_a + s_a s_o^2 - s_o^3 - (3 a S_A)/2

X896

-1/2 a b c - a^2 s_a + s_abc + (a b + a c - b c) s_o + s_a s_o^2 - (7 a S_A)/2

asymmetrize2[a * (2 * a^2 - b^2 - c^2)//Expand]

-1/2 a b c - a^2 s_a - s_a s_o^2 + s_o^3 - (5 a S_A)/2

X908

asymmetrize[a^2 * b - b^3 + a^2 * c - 2 * a * b * c + b^2 * c + b * c^2 - c^3//Expand]

X918

asymmetrize[(b - c) * (-(a * b) + b^2 - a * c + c^2)//Expand]

X1150

asymmetrize[a^3 - a * b^2 - b^2 * c - a * c^2 - b * c^2//Expand]

X1155

asymmetrize[a * (2 * a^2 - a * b - b^2 - a * c + 2 * b * c - c^2)//Expand]

X1376

asymmetrize[a * (a^2 - a * b - a * c + 2 * b * c)//Expand]

X1621

asymmetrize[a * (a^2 - a * b - a * c - b * c)//Expand]

X1699

asymmetrize[a^3 + a * b^2 - 2 * b^3 - 2 * a * b * c + 2 * b^2 * c + a * c^2 + 2 * b * c^2 - 2 * c^3//Expand]

X1836

asymmetrize[a^3 - b^3 + b^2 * c + b * c^2 - c^3//Expand]

X1997

asymmetrize[a^3 - a^2 * b - a * b^2 + b^3 - a^2 * c + 8 * a * b * c - 3 * b^2 * c - a * c^2 - 3 * b * c^2 + c^3//Expand]

X1999

asymmetrize[a^3 + a^2 * b + a^2 * c + a * b * c - b^2 * c - b * c^2//Expand]

X2094

asymmetrize[5 * a^3 + a^2 * b - 5 * a * b^2 - b^3 + a^2 * c + 6 * a * b * c + b^2 * c - 5 * a * c^2 + b * c^2 - c^3//Expand]

X2550

asymmetrize[a^3 - a^2 * b + a * b^2 - b^3 - a^2 * c + 2 * a * b * c + b^2 * c + a * c^2 + b * c^2 - c^3//Expand]

Created by Mathematica (September 26, 2005)