The shape parameter s_abc/abc

(not completed) The ratio of symmetric functions determines the positioning of centers on lines. I choose a more general definition of center than the usual one. If a points coordinates add to a symmetric function in triangle parameters, it is a center as far as I am concerned. Hence I include the Brocard points as well as the symmedian point as a center.
Let the coordinartes of a point be c, and their sum T.  The normalized point P is then C/T. If C3 = C1 + C2, then T3 = T1 + T2, so that the normalized form for P3 can be written

P3 = T1/(T1 + T2) P1 + T2/(T1 + T2) P2 . (1)

T1 and T2 thus determine the relative position of P3 on the line P1—P2. More precisely  the ratio P3—P2/(P3 - P1)= T1/T2 determines the position of the point. This makes the symmetric functions of triangle parameters very important indeed. Lists of symmetric functions are found below, both for weak and strong structures.

As an example of this technique let us find the position of pK = : b^4: on the OK line. We begin with the following trick which lets use express this point in terms of the coordinates for O and for K. :b^4: = :b^2S_Ω-b^2S_B:. Now we set :b^2: = S_Ω K and :b^2S_B: = 2S^2 O, where O and K are vectors that represent the points. Hence

pK = (S_Ω^2K - 2S^2 O)/(S_Ω^2 - 2S^2) which means that pK is on the K side of the OK line and is determined by the ratio 2S^2/S_Ω^2= 2 tan^2Ω.

Weak symmetric functions

first degree s_o = 1/2 (a + b + c)
second degree S_Ω = 1/2 (a^2 + b^2 + c^2) = S cot Ω
□ S_11 = a b + b c + c a = 2 s_o^2 - S_Ω
third degree S_111 = a b c
□ S_3 = a^3 + b^3 + c^3 = s_o^3 - 3s_oS_11 + 6S_111 = -5s_o^3 + 3s_oS_Ω + 6a b c
□ s_abc
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Strong Identities

□ S = 2Area
second degree S_Ω = 1/2 (a^2 + b^2 + c^2) = S cot Ω
fourth degree S_22 = b^2c^2 + c^2a^2 + a^2b^2 = S_Ω^2 + S^2
□ S_4 = a^4 + b^4 + c^4 = 2S_Ω^2 - 2S^2
□ S^2 = 2b^2c^2 + 2c^2a^2 + 2a^2b^2 - a^4 - b^4 - c^4
sixth degree S_222 = a^2b^2c^2 = S^2S_Ω - S_ABC
□ S_42 = S_22S_2 - 3S_222 = 2S_Ω^3 - S^2S_Ω + 3S_ABC
□ S_6 = 2S_Ω^3 - 3S^2S_Ω - 3S_ABC
eighth degree S_422 = 2S^2S_Ω^2 - 2S_ABCΩ
□ S_44 = S_Ω^4 - 2S^2S_Ω^2 + S^4 + 4S_ABCΩ
□ S_62 = 2S_Ω^4 - 2S^2S_Ω^2 - 2S^4 + 2S_ABCΩ
□ S_8 = 2S_Ω^4 - 4S^2S_Ω^2 + 2S^4 + 8S_ABCΩ
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identities relevant to point coordinates

∑ a = 2s_o ∑ s_a = s_o
∑a b = 2s_o^2 - S_Ω ∑s_ab = s_o^2 - S_Ω
∑a s_a = 2s_o^2 - 2S_Ω ∑a s_bc = abc - 2s_abc
∑a S_A = abc + 4s_abc abc
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s_ob + s_ca = ca s_ob - s_ca = S_B
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∑a^2 = S_Ω ∑b^2c^2 = S_Ω^2 + S^2
∑S_A = S_Ω ∑S_BC = S^2
□ ∑a^2S_A = 2S^2
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∑a^(4 -) = S_Ω^2 + 3S^2 ∑a^(4 +) = 3S_Ω^2 - S^2
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