Symmetrizing point coordinates

There are symmetrizations of the 2nd degree points in Brisse's collection. 3rd degree are here. It is possible for the algebra of geometry to correspond to that geometry itself. Since conceptions of center matter so much geometrically, their algebraic correlate, algebraic symmetry, is to be desired in our formulas. When it is present the formulas mean more and we learn more from them. I have written special routines in Mathmatica to do this.

X142

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

-2 a s_a - 4 s_bc

X144

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

-4 a s_a + 4 s_bc

X192 but not simpler

This point is best left with the coordinates it has since it is a simple relation of the symmetric form ab+bc+ca.

asymmetrize[-(a * b) - a * c + b * c]//.antirules

-2 a s_a + 2 s_oa - S_Ω

X239 not simpler

asymmetrize[a^2 - b * c]//Simplify

-2 s_oa + S_Ω

X319

asymmetrize[a^2 - b^2 - b * c - c^2]//Simplify

s_bc - 3 s_oa

X320

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

3 s_bc - s_oa

X344

1/2 (asymmetrize[a^2 - 2 * a * b + b^2 - 2 * a * c + c^2]//Simplify)//.antirules

-2 a s_a + S_A

X(527)

(1/2 asymmetrize[2 * a^2 - a * b - b^2 - a * c + 2 * b * c - c^2]//Simplify)//.antirules

-a s_a + 2 s_bc

X536

(1/2 asymmetrize[-(a * b) - a * c + 2 * b * c]//Simplify)//.antirules

1/2 (-2 a s_a + s_bc + 3 s_oa - S_Ω)

X545

In[646]:=

( asymmetrize[2 * a^2 - 2 * a * b - b^2 - 2 * a * c + 4 * b * c - c^2]//Simplify)//.antirules

Out[646]=

-4 a s_a + 5 s_bc + 3 s_oa - S_Ω

X966

( -1/2asymmetrize[a^2 - 2 * a * b - b^2 - 2 * a * c - 2 * b * c - c^2]//Simplify)//.antirules

b c + 2 a s_a + S_Ω

X1266

( -1/2asymmetrize[a * b + b^2 + a * c - 4 * b * c + c^2]//Simplify)//.antirules

-a s_a + 2 s_bc + 2 s_oa - S_Ω

= -a s_a+2 bc-S_Ω

X1278

( -1/2asymmetrize[-(a * b) - a * c + 3 * b * c]//Simplify)//.antirules

a s_a - s_bc - 2 s_oa + S_Ω/2

X1654

( -1/2asymmetrize[a^2 - a * b - b^2 - a * c - b * c - c^2]//Simplify)//.antirules

a s_a + s_oa + S_Ω/2

X1743

( -1/2asymmetrize[a * (3 * a - b - c)//Expand]//Simplify)//.antirules

a s_a + S_A - S_Ω

= a s_a- a^2

X1992

( -1/2asymmetrize[5 * a^2 - b^2 - c^2]//Simplify)//.antirules

-3 s_bc + 3 s_oa - 2 S_Ω

X2345

( 1/2asymmetrize[a^2 + b^2 + 2 * b * c + c^2]//Simplify)//.antirules

b c + S_Ω

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