Weakened Isodynamic points
an example of octile desmic structure

updated 5/15/06

Points that come in two versions, such as the Fermat and isodynamic points, are almost always on strong lines. Theory tells us this and ETC verifies it empirically.

I have been investigating triangle geometry from the point of view of the algebraic structures as well as trying to "mine the Kimberling points" found in ETC. Both of these methods have been very fertile. This page combines both investigations.

The Fermat and isodynamic points have coordinates related to strong points and each pair is on a strong line. The coordinates contain the factor S cot π/3, where S is twice the triangle area. This factor contains two square roots, which algebraically isolates them from the great mass of triangle points. The coordinates of the isodynamic points are (in barycentrics for fundamental structural reasons) ( : b² S_B ± b² S/√3 : ).

The isodynamic points are on the OK line. The weakened OK line is the M_o—I_o line, the Mittenpunkt-Incenter line. So the coordinates of the weakened isdynamic point should look like ( : b s_b ± b Z : ) where Z is an irrational factor.

There is no reason that these types of points could not be on weak lines. In fact X(1251) = g X(1082) in ETC is on a weak line as well as the Feuerbach hyperbola, itself weak. These types of points have never been researched as far as I can tell.

With X1082 = : b s_ca + b Z: we should consider : b s_ob + b Z : .

Remember there are 8 versions of each of these points. Here is a picture of then all, together with the lines thereon. Other octile points can be found on the Simmons conics. The points listed in the picture are found in the table below.

Figure: This shows the two sets of octile points (green and yellow) on the I_x—M_x (x = o, a, b, c) lines, which concur at K; the O—I_x lines; and the M_x—W_x lines, where the W_x (cyan) are a type of Clawson point.

Let's make some sense of them. So lets consider all 8 versions. Give the points two indices, one for each type of extraversion (n,s) and (o,a,b,c). Define these point as

S_no = ( : b s_bc + b Z : )
R_no = (: b s_ob + b Z : )

The S points (green) are on O—I_o; the R points (yellow) on K—I_o.

Constructing these points

From ETC we find all the information we need to construct one of these points, which gives the key to construct them all. The following picture shows the construction of S_nc and S_sc. In this picture F_n and F_s are the two Fermat points, A_Gc is the A-vertex of the Cevian triangle of G_c. H_Gc is the orthocenter of this triangle, and P/Q = G_c/H_Gc is the Cevian quotient listed in the construction

Adding and subtracting all combinations we get the following points sharing lines with these points. Here W_o = : b S_B : is one of the Claswon points.

G, W_o, I_o, abc ± a Z, b S_B ± a Z

The lower left part of this chart shows the result of adding the two coordinates together. The upper right is subtracted.

	:b sob + bZ	:b sob – bZ	:b sca + bZ	:b sca – bZ	–
:b sob + bZ:	–	Io	Wo	: b SB + 2aZ:	:b sob + bZ:
:b sob – bZ:	Mo	–	: b SB – 2aZ:	Wo	:b sob – bZ:
:b sca + bZ:	: abc + 2aZ:	G	–	Io	:b sca + bZ:
: b sca – bZ:	G	: abc + 2aZ:	gMo	–	: b sca – bZ:
+	:b sob + bZ	:b sob – bZ:	:b sca + bZ:	:b sca – bZ:

Here are the four O—I_x lines.

Here are the points on the four K—I_x lions.

Here are the G—sx—nx lines

Desmic Systems

The extraversions of the R points are

R_no = ( a s_oa – a Z : b s_ob – b Z : c s_oc – c Z )
R_na = ( a s_oa – a Z : b s_ca + b Z : c s_ab + c Z )
R_nb = ( a s_bc + a Z : b s_ob – b Z , c s_ab + c Z )

Note that extraversions of the R points contain coordinates of the S points, which gives us this desmic system, illustrated below the table

Desmic system
G	A	B	C
Rno	Rna	Rnb	Rnc
Sso	Ssa	Ssb	Ssc
Harmon: (:bSB + 2b Z:)

The Harmon represents another set of octile points which fit in pairs on the W_x—I_x lines.

There is another one where n and s are reversed.

Desmic system
G	A	B	C
Rso	Rsa	Rsb	Rsc
Sno	Sna	Snb	Snc
Harmon: (:bSB – 2b Z:)