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X(523), X(2489), X(23285), X(31998), X(36953)

vertices of the medial triangle

S1 = X(523)^4

more points below

Geometric properties :

K1152 is an example of trident as in CL070 where general properties can be found. Its pseudo-pole is X(8029), the barycentric cube of X(523).

K1152 is globally invariant under the two transformations 𝛗₁ and 𝛗₂ mentioned in CL070.

It can be parametrized as follows. Let U = a^2 / (- a^2 + b^2 + c^2) and define V, W cyclically.

Then P(T) = (b^2 - c^2) [1 - (T - 1) U] / (T - 2 U) : : , is a point on K1152 for any real number T or infinity.

Furthermore, 𝛗₁(P(T)) = P(2 / (T - 1)) and 𝛗₂(P(T)) = P(1 + 2 / T). Recall that 𝛗₁ and 𝛗₂ are inverse of one another.

In particular, P(∞) = X(2489), P(-1) = P(2) = X(523).

Hence, starting from one point P(T) on K1152 which is not X(523), an infinite sequence of points of K1152 can be defined by applying 𝛗₁ in one direction (from left to right in the table below) and 𝛗₂ in the opposite direction.

One sequence of points on K1152 is particularly remarkable, that starting at X(2489) = P(∞).

S2 = (b - c) (b + c) (a^2 + b^2 - 3 c^2) (-a^2 + 3 b^2 - c^2) (-a^2 + b^2 + c^2) : : , SEARCH = 1.82689909607982

S3 = (b - c) (b + c) (3 a^2 + 3 b^2 - 5 c^2) (-3 a^2 + 5 b^2 - 3 c^2) (-3 a^2 + b^2 + c^2) : : , SEARCH = 2.12142254848522

S5 = (a^2 + b^2) (b - c) (b + c) (a^2 + c^2) (2 a^2 + b^2 + c^2) : : , SEARCH = 17.7954170190205

S6 = (b - c) (b + c) (a^2 + 2 b^2 + c^2) (a^2 + b^2 + 2 c^2) (2 a^2 + 3 b^2 + 3 c^2) : : , SEARCH = 158.136403729115

S7 = (b-c) (b+c) (5 a^2+5 b^2-11 c^2) (5 a^2-3 b^2-3 c^2) (5 a^2-11 b^2+5 c^2) : : , SEARCH = -3.50333512408298

S8 = (b-c) (b+c) (3 a^2+3 b^2+2 c^2) (3 a^2+2 b^2+3 c^2) (6 a^2+5 b^2+5 c^2) : : , SEARCH = 452.091971784400

As we move along in the table towards the right (resp. the left), T approaches -1 (resp. 2) and P(T) goes to the infinite point X(523).

Recall that S1 = X(523)^4, SEARCH = 1.25486105681575 and 𝛗₂(S1) = X(36953), 𝛗₁(S1) = X(31998) then 𝛗₁(X(31998)) = S4, SEARCH = 12.6352166218187.

S4 = (b-c) (b+c) (-a^4-a^2 b^2+b^4+3 a^2 c^2-b^2 c^2-c^4) (a^4-3 a^2 b^2+b^4+a^2 c^2+b^2 c^2-c^4) (-a^4+a^2 b^2+3 b^4+a^2 c^2-7 b^2 c^2+3 c^4) : : .

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All these properties are easily generalized and adapted when X(523) = b^2 - c^2 : c^2 - a^2 : a^2 - b^2 is replaced by the infinite point P = v - w : w - u : u -w , giving another trident K(P). See other examples in CL070.