Sharing centroid and more

 We study  orthology and perspectivity in a simple configuration. We find two sister cubics and two points on Euler line that are not catalogued yet.

Download: Sharing the centroid and more

For convenience we give here in text format the first coordinate and lines through them of new points on Euler line.

Point E_1 on cubic Gamma_1: 

(a^2 + b^2 + c^2) (2 a^4 - a^2 b^2 - b^4 - a^2 c^2 + 2 b^2 c^2 -  c^4) + (3 a^4 - 2 a^2 b^2 - b^4 - 2 a^2 c^2 + 2 b^2 c^2 - c^4) S

on lines {2, 3}, {511, 39888}, {637, 9873}, {1503, 49038}, {1975,   58803}, {5490, 54935}, {5870, 9733}, {6201, 43119}, {6459,   31670}, {6460, 39876}, {6560, 7738}, {7690, 33364}, {7750,   58804}, {8982, 14927}, {10722, 33341}, {10784, 45488}, {26361,   45542}, {26429, 42413}, {26441, 51212}, {29181, 49039}, {29317,   42858}, {39874, 43133}, {41411, 42275}, {42258, 48910}, {42259,  48905}, {42264, 63548}

Point E_2 on cubic Gamma_2: 

(a^2 + b^2 + c^2) (2 a^4 - a^2 b^2 - b^4 - a^2 c^2 + 2 b^2 c^2 -     c^4) + (-3 a^4 + 2 a^2 b^2 + b^4 + 2 a^2 c^2 - 2 b^2 c^2 + c^4) S

on lines {2, 3}, {511, 39887}, {638, 9873}, {1503, 49039}, {1975,   58804}, {5491, 54936}, {5871, 9732}, {6202, 43118}, {6459,   39875}, {6460, 31670}, {6561, 7738}, {7692, 33365}, {7750,   58803}, {8982, 51212}, {10722, 33340}, {10783, 45489}, {14927,  26441}, 26362, 45543}, {26430, 42414}, {29181, 49038}, {29317,   42859}, {39874, 43134}, {41410, 42276}, {42258, 48905}, {42259,  48910}, {42263, 63548}