[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
The Euler line of ABC intersects BC, CA, AB at A1, B1, C1, resp.
The Euler line of the orthic triangle intersects BC, CA, AB at A2, B2, C2, resp.
The Euler lines of NA1A2, NB1B2, NC1C2 bound a triangle A*B*C*.
ABC, A*B*C* are parallelogic.
The Euler line of ABC intersects BC, CA, AB at A1, B1, C1, resp.
The Euler line of the orthic triangle intersects BC, CA, AB at A2, B2, C2, resp.
The Euler lines of NA1A2, NB1B2, NC1C2 bound a triangle A*B*C*.
ABC, A*B*C* are parallelogic.
[Peter Moses]:
Hi Antreas,
A nice point ...
b^2 c^2 (a^2-a b+b^2-c^2)^2 (a^2+a b+b^2-c^2)^2 (-a^2+b^2-a c-c^2)^2 (-a^2+b^2+a c-c^2)^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (-a^4+a^2 b^2+2 a^2 c^2+b^2 c^2-c^4)::
on line {476,1141}.
X(51)-cross conjugate of X(1989).
isoconjugate of X(j) and X(j) for these (i,j): {{323, 2290}, {1154, 6149}}.
cevapoint of X(51) and X(1989).
barycentric product X(i)X(j) for these {i,j}: {{94, 1141}, {276, 14595}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{94, 1273}, {1141, 323}, {1989, 1154}, {8882, 3043}, {14595, 216}}.
and a terrible one ...
(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^32 b^4-12 a^30 b^6+64 a^28 b^8-196 a^26 b^10+363 a^24 b^12-352 a^22 b^14-66 a^20 b^16+792 a^18 b^18-1353 a^16 b^20+1364 a^14 b^22-924 a^12 b^24+428 a^10 b^26-131 a^8 b^28+24 a^6 b^30-2 a^4 b^32+2 a^32 b^2 c^2-21 a^30 b^4 c^2+101 a^28 b^6 c^2-292 a^26 b^8 c^2+563 a^24 b^10 c^2-779 a^22 b^12 c^2+882 a^20 b^14 c^2-1032 a^18 b^16 c^2+1344 a^16 b^18 c^2-1571 a^14 b^20 c^2+1377 a^12 b^22 c^2-836 a^10 b^24 c^2+331 a^8 b^26 c^2-77 a^6 b^28 c^2+8 a^4 b^30 c^2+a^32 c^4-21 a^30 b^2 c^4+126 a^28 b^4 c^4-380 a^26 b^6 c^4+683 a^24 b^8 c^4-780 a^22 b^10 c^4+577 a^20 b^12 c^4-285 a^18 b^14 c^4+99 a^16 b^16 c^4+35 a^14 b^18 c^4-180 a^12 b^20 c^4+234 a^10 b^22 c^4-151 a^8 b^24 c^4+46 a^6 b^26 c^4-3 a^4 b^28 c^4-a^2 b^30 c^4-12 a^30 c^6+101 a^28 b^2 c^6-380 a^26 b^4 c^6+814 a^24 b^6 c^6-1071 a^22 b^8 c^6+872 a^20 b^10 c^6-358 a^18 b^12 c^6-162 a^16 b^14 c^6+489 a^14 b^16 c^6-536 a^12 b^18 c^6+391 a^10 b^20 c^6-223 a^8 b^22 c^6+102 a^6 b^24 c^6-33 a^4 b^26 c^6+7 a^2 b^28 c^6-b^30 c^6+64 a^28 c^8-292 a^26 b^2 c^8+683 a^24 b^4 c^8-1071 a^22 b^6 c^8+1126 a^20 b^8 c^8-785 a^18 b^10 c^8+435 a^16 b^12 c^8-246 a^14 b^14 c^8+107 a^12 b^16 c^8-115 a^10 b^18 c^8+234 a^8 b^20 c^8-185 a^6 b^22 c^8+51 a^4 b^24 c^8-18 a^2 b^26 c^8+12 b^28 c^8-196 a^26 c^10+563 a^24 b^2 c^10-780 a^22 b^4 c^10+872 a^20 b^6 c^10-785 a^18 b^8 c^10+390 a^16 b^10 c^10-152 a^14 b^12 c^10+320 a^12 b^14 c^10-252 a^10 b^16 c^10-154 a^8 b^18 c^10+151 a^6 b^20 c^10+75 a^4 b^22 c^10+14 a^2 b^24 c^10-66 b^26 c^10+363 a^24 c^12-779 a^22 b^2 c^12+577 a^20 b^4 c^12-358 a^18 b^6 c^12+435 a^16 b^8 c^12-152 a^14 b^10 c^12-328 a^12 b^12 c^12+150 a^10 b^14 c^12+356 a^8 b^16 c^12-108 a^6 b^18 c^12-401 a^4 b^20 c^12+25 a^2 b^22 c^12+220 b^24 c^12-352 a^22 c^14+882 a^20 b^2 c^14-285 a^18 b^4 c^14-162 a^16 b^6 c^14-246 a^14 b^8 c^14+320 a^12 b^10 c^14+150 a^10 b^12 c^14-524 a^8 b^14 c^14+47 a^6 b^16 c^14+782 a^4 b^18 c^14-63 a^2 b^20 c^14-495 b^22 c^14-66 a^20 c^16-1032 a^18 b^2 c^16+99 a^16 b^4 c^16+489 a^14 b^6 c^16+107 a^12 b^8 c^16-252 a^10 b^10 c^16+356 a^8 b^12 c^16+47 a^6 b^14 c^16-954 a^4 b^16 c^16+36 a^2 b^18 c^16+792 b^20 c^16+792 a^18 c^18+1344 a^16 b^2 c^18+35 a^14 b^4 c^18-536 a^12 b^6 c^18-115 a^10 b^8 c^18-154 a^8 b^10 c^18-108 a^6 b^12 c^18+782 a^4 b^14 c^18+36 a^2 b^16 c^18-924 b^18 c^18-1353 a^16 c^20-1571 a^14 b^2 c^20-180 a^12 b^4 c^20+391 a^10 b^6 c^20+234 a^8 b^8 c^20+151 a^6 b^10 c^20-401 a^4 b^12 c^20-63 a^2 b^14 c^20+792 b^16 c^20+1364 a^14 c^22+1377 a^12 b^2 c^22+234 a^10 b^4 c^22-223 a^8 b^6 c^22-185 a^6 b^8 c^22+75 a^4 b^10 c^22+25 a^2 b^12 c^22-495 b^14 c^22-924 a^12 c^24-836 a^10 b^2 c^24-151 a^8 b^4 c^24+102 a^6 b^6 c^24+51 a^4 b^8 c^24+14 a^2 b^10 c^24+220 b^12 c^24+428 a^10 c^26+331 a^8 b^2 c^26+46 a^6 b^4 c^26-33 a^4 b^6 c^26-18 a^2 b^8 c^26-66 b^10 c^26-131 a^8 c^28-77 a^6 b^2 c^28-3 a^4 b^4 c^28+7 a^2 b^6 c^28+12 b^8 c^28+24 a^6 c^30+8 a^4 b^2 c^30-a^2 b^4 c^30-b^6 c^30-2 a^4 c^32)::
X(51)-cross conjugate of X(1989).
isoconjugate of X(j) and X(j) for these (i,j): {{323, 2290}, {1154, 6149}}.
cevapoint of X(51) and X(1989).
barycentric product X(i)X(j) for these {i,j}: {{94, 1141}, {276, 14595}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{94, 1273}, {1141, 323}, {1989, 1154}, {8882, 3043}, {14595, 216}}.
and a terrible one ...
(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^32 b^4-12 a^30 b^6+64 a^28 b^8-196 a^26 b^10+363 a^24 b^12-352 a^22 b^14-66 a^20 b^16+792 a^18 b^18-1353 a^16 b^20+1364 a^14 b^22-924 a^12 b^24+428 a^10 b^26-131 a^8 b^28+24 a^6 b^30-2 a^4 b^32+2 a^32 b^2 c^2-21 a^30 b^4 c^2+101 a^28 b^6 c^2-292 a^26 b^8 c^2+563 a^24 b^10 c^2-779 a^22 b^12 c^2+882 a^20 b^14 c^2-1032 a^18 b^16 c^2+1344 a^16 b^18 c^2-1571 a^14 b^20 c^2+1377 a^12 b^22 c^2-836 a^10 b^24 c^2+331 a^8 b^26 c^2-77 a^6 b^28 c^2+8 a^4 b^30 c^2+a^32 c^4-21 a^30 b^2 c^4+126 a^28 b^4 c^4-380 a^26 b^6 c^4+683 a^24 b^8 c^4-780 a^22 b^10 c^4+577 a^20 b^12 c^4-285 a^18 b^14 c^4+99 a^16 b^16 c^4+35 a^14 b^18 c^4-180 a^12 b^20 c^4+234 a^10 b^22 c^4-151 a^8 b^24 c^4+46 a^6 b^26 c^4-3 a^4 b^28 c^4-a^2 b^30 c^4-12 a^30 c^6+101 a^28 b^2 c^6-380 a^26 b^4 c^6+814 a^24 b^6 c^6-1071 a^22 b^8 c^6+872 a^20 b^10 c^6-358 a^18 b^12 c^6-162 a^16 b^14 c^6+489 a^14 b^16 c^6-536 a^12 b^18 c^6+391 a^10 b^20 c^6-223 a^8 b^22 c^6+102 a^6 b^24 c^6-33 a^4 b^26 c^6+7 a^2 b^28 c^6-b^30 c^6+64 a^28 c^8-292 a^26 b^2 c^8+683 a^24 b^4 c^8-1071 a^22 b^6 c^8+1126 a^20 b^8 c^8-785 a^18 b^10 c^8+435 a^16 b^12 c^8-246 a^14 b^14 c^8+107 a^12 b^16 c^8-115 a^10 b^18 c^8+234 a^8 b^20 c^8-185 a^6 b^22 c^8+51 a^4 b^24 c^8-18 a^2 b^26 c^8+12 b^28 c^8-196 a^26 c^10+563 a^24 b^2 c^10-780 a^22 b^4 c^10+872 a^20 b^6 c^10-785 a^18 b^8 c^10+390 a^16 b^10 c^10-152 a^14 b^12 c^10+320 a^12 b^14 c^10-252 a^10 b^16 c^10-154 a^8 b^18 c^10+151 a^6 b^20 c^10+75 a^4 b^22 c^10+14 a^2 b^24 c^10-66 b^26 c^10+363 a^24 c^12-779 a^22 b^2 c^12+577 a^20 b^4 c^12-358 a^18 b^6 c^12+435 a^16 b^8 c^12-152 a^14 b^10 c^12-328 a^12 b^12 c^12+150 a^10 b^14 c^12+356 a^8 b^16 c^12-108 a^6 b^18 c^12-401 a^4 b^20 c^12+25 a^2 b^22 c^12+220 b^24 c^12-352 a^22 c^14+882 a^20 b^2 c^14-285 a^18 b^4 c^14-162 a^16 b^6 c^14-246 a^14 b^8 c^14+320 a^12 b^10 c^14+150 a^10 b^12 c^14-524 a^8 b^14 c^14+47 a^6 b^16 c^14+782 a^4 b^18 c^14-63 a^2 b^20 c^14-495 b^22 c^14-66 a^20 c^16-1032 a^18 b^2 c^16+99 a^16 b^4 c^16+489 a^14 b^6 c^16+107 a^12 b^8 c^16-252 a^10 b^10 c^16+356 a^8 b^12 c^16+47 a^6 b^14 c^16-954 a^4 b^16 c^16+36 a^2 b^18 c^16+792 b^20 c^16+792 a^18 c^18+1344 a^16 b^2 c^18+35 a^14 b^4 c^18-536 a^12 b^6 c^18-115 a^10 b^8 c^18-154 a^8 b^10 c^18-108 a^6 b^12 c^18+782 a^4 b^14 c^18+36 a^2 b^16 c^18-924 b^18 c^18-1353 a^16 c^20-1571 a^14 b^2 c^20-180 a^12 b^4 c^20+391 a^10 b^6 c^20+234 a^8 b^8 c^20+151 a^6 b^10 c^20-401 a^4 b^12 c^20-63 a^2 b^14 c^20+792 b^16 c^20+1364 a^14 c^22+1377 a^12 b^2 c^22+234 a^10 b^4 c^22-223 a^8 b^6 c^22-185 a^6 b^8 c^22+75 a^4 b^10 c^22+25 a^2 b^12 c^22-495 b^14 c^22-924 a^12 c^24-836 a^10 b^2 c^24-151 a^8 b^4 c^24+102 a^6 b^6 c^24+51 a^4 b^8 c^24+14 a^2 b^10 c^24+220 b^12 c^24+428 a^10 c^26+331 a^8 b^2 c^26+46 a^6 b^4 c^26-33 a^4 b^6 c^26-18 a^2 b^8 c^26-66 b^10 c^26-131 a^8 c^28-77 a^6 b^2 c^28-3 a^4 b^4 c^28+7 a^2 b^6 c^28+12 b^8 c^28+24 a^6 c^30+8 a^4 b^2 c^30-a^2 b^4 c^30-b^6 c^30-2 a^4 c^32)::
on no lines.
Best regards,
Peter Moses.
Best regards,
Peter Moses.
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