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X(3), X(30), X(1379), X(1380) X(39162), X(39163), X(39164), X(39165) : foci of the Steiner inellipse and their reflections in X(3) other points below |
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Geometric properties : |
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K1294 is the Hessian of the central stelloid K100. K1294 is a central focal cubic with center and singular focus O, and inflexional tangent passing through X(74) and X(110). The real asymptote is the Euler line. K1294 meets the axes of the inscribed Steiner ellipse again at two points R1 and R2 obviously symmetric about O, on the line {3,67} and on the circle C(O, G). Their barycentric product P lies on the line {3,524}. P = 5 a^8-9 a^6 b^2+5 a^4 b^4-b^8-9 a^6 c^2+11 a^4 b^2 c^2-4 a^2 b^4 c^2+5 a^4 c^4-4 a^2 b^2 c^4+2 b^4 c^4-c^8 : : , SEARCH = 13.9074017379856. R1 = 2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6+6 a^2 SA Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4] : : , SEARCH = 3.71108330514737, on the lines {2,1341}, {3,67}, {30,1380}, {376,3414}, {524,1379}. R2 = 2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6-6 a^2 SA Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4] : : , SEARCH = 9.85364248324532, on the lines {2,1340}, {3,67}, {30,1379}, {376,3413}, {524,1380}. See the related cubics K706, K758, K1293. K1294 belongs to the pencil of focal cubics mentioned in Table 48 which contains K038, K187, K463, K800. ***
K1294 is invariant under an involution very similar to the Psi involution mentioned in page K018. For every point M on K1294, the polar conic of M with respect to the stelloid K100 is the union of two perpendicular lines secant at N which is the conjugate of M in K1294. The mapping F : M → N is a quadratic involution which globally leaves K1294 invariant. Let (L) be the line passing through {3,2575} and (C) be the circle with center O, orthogonal to the circles with diameters {2,1113} and {376,1114}. F is the commutative product of the reflection in (L) and the inversion in (C). F swaps the pairs of foci of the Steiner inscribed ellipse and the pairs of points {X1379,R1}, {X1380,R2}. The Euler line (asymptote) is transformed into the line {3,74} (inflexional tangent), and the Brocard axis into the line {3,67}. The circumcircle is transformed into the circle C(O,G) and the Brocard circle into the line {351,690}. With M = u:v:w, N = F(M) = a^2(c^2 (2 a^2-b^2-2 c^2) u v+b^2 (2 a^2-2 b^2-c^2) u w+(-a^2+b^2+c^2) (c^2 v^2+b^2 w^2)+b^2c^2u^2+(-3 a^2+b^2+c^2) (-a^2+b^2+c^2) v w) : : . |