   too complicated to be written here. Click on the link to download a text file.  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(∞). P(T) S7 S3 S2 X(2489) X(23285) S5 S6 S8 T 21/11 11/5 5/3 3 1 ∞ 0 -2 -2/3 -6/5 -10/11   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) : : . *** 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.  