[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
Na, Oa = the midpoints of NA, OA, resp.
N1, O1 = the reflections of Na, Oa in OA, NA, resp.
A' = NaOa /\ N1O1
Hi Antreas,
A very long-winded point on Euler line:
3 a^36 b^4-39 a^34 b^6+228 a^32 b^8-780 a^30 b^10+1680 a^28 b^12-2184 a^26 b^14+1092 a^24 b^16+1716 a^22 b^18-4290 a^20 b^20+4290 a^18 b^22-1716 a^16 b^24-1092 a^14 b^26+2184 a^12 b^28-1680 a^10 b^30+780 a^8 b^32-228 a^6 b^34+39 a^4 b^36-3 a^2 b^38+10 a^36 b^2 c^2-145 a^34 b^4 c^2+882 a^32 b^6 c^2-2986 a^30 b^8 c^2+6141 a^28 b^10 c^2-7622 a^26 b^12 c^2+4965 a^24 b^14 c^2-866 a^22 b^16 c^2+869 a^20 b^18 c^2-4312 a^18 b^20 c^2+3949 a^16 b^22 c^2+2850 a^14 b^24 c^2-9689 a^12 b^26 c^2+10470 a^10 b^28 c^2-6529 a^8 b^30 c^2+2538 a^6 b^32 c^2-595 a^4 b^34 c^2+73 a^2 b^36 c^2-3 b^38 c^2+3 a^36 c^4-145 a^34 b^2 c^4+1316 a^32 b^4 c^4-5418 a^30 b^6 c^4+12216 a^28 b^8 c^4-15408 a^26 b^10 c^4+8907 a^24 b^12 c^4+1975 a^22 b^14 c^4-7463 a^20 b^16 c^4+8577 a^18 b^18 c^4-9164 a^16 b^20 c^4+3100 a^14 b^22 c^4+12006 a^12 b^24 c^4-23586 a^10 b^26 c^4+21507 a^8 b^28 c^4-11481 a^6 b^30 c^4+3638 a^4 b^32 c^4-622 a^2 b^34 c^4+42 b^36 c^4-39 a^34 c^6+882 a^32 b^2 c^6-5418 a^30 b^4 c^6+15158 a^28 b^6 c^6-21282 a^26 b^8 c^6+12037 a^24 b^10 c^6+4905 a^22 b^12 c^6-11061 a^20 b^14 c^6+4875 a^18 b^16 c^6+4971 a^16 b^18 c^6-11454 a^14 b^20 c^6+3138 a^12 b^22 c^6+19666 a^10 b^24 c^6-34263 a^8 b^26 c^6+27431 a^6 b^28 c^6-12099 a^4 b^30 c^6+2820 a^2 b^32 c^6-267 b^34 c^6+228 a^32 c^8-2986 a^30 b^2 c^8+12216 a^28 b^4 c^8-21282 a^26 b^6 c^8+13390 a^24 b^8 c^8+5662 a^22 b^10 c^8-10450 a^20 b^12 c^8+472 a^18 b^14 c^8+5425 a^16 b^16 c^8+949 a^14 b^18 c^8-10747 a^12 b^20 c^8+2473 a^10 b^22 c^8+23737 a^8 b^24 c^8-36453 a^6 b^26 c^8+24201 a^4 b^28 c^8-7843 a^2 b^30 c^8+1008 b^32 c^8-780 a^30 c^10+6141 a^28 b^2 c^10-15408 a^26 b^4 c^10+12037 a^24 b^6 c^10+5662 a^22 b^8 c^10-10298 a^20 b^10 c^10+266 a^18 b^12 c^10+2380 a^16 b^14 c^10+2179 a^14 b^16 c^10+1103 a^12 b^18 c^10-10625 a^10 b^20 c^10+1339 a^8 b^22 c^10+23863 a^6 b^24 c^10-29442 a^4 b^26 c^10+14043 a^2 b^28 c^10-2460 b^30 c^10+1680 a^28 c^12-7622 a^26 b^2 c^12+8907 a^24 b^4 c^12+4905 a^22 b^6 c^12-10450 a^20 b^8 c^12+266 a^18 b^10 c^12+2758 a^16 b^12 c^12-996 a^14 b^14 c^12+2557 a^12 b^16 c^12+1337 a^10 b^18 c^12-10255 a^8 b^20 c^12-843 a^6 b^22 c^12+19803 a^4 b^24 c^12-15911 a^2 b^26 c^12+3864 b^28 c^12-2184 a^26 c^14+4965 a^24 b^2 c^14+1975 a^22 b^4 c^14-11061 a^20 b^6 c^14+472 a^18 b^8 c^14+2380 a^16 b^10 c^14-996 a^14 b^12 c^14-1104 a^12 b^14 c^14+1945 a^10 b^16 c^14+1501 a^8 b^18 c^14-9339 a^6 b^20 c^14-3821 a^4 b^22 c^14+10047 a^2 b^24 c^14-3276 b^26 c^14+1092 a^24 c^16-866 a^22 b^2 c^16-7463 a^20 b^4 c^16+4875 a^18 b^6 c^16+5425 a^16 b^8 c^16+2179 a^14 b^10 c^16+2557 a^12 b^12 c^16+1945 a^10 b^14 c^16+4366 a^8 b^16 c^16+4512 a^6 b^18 c^16-4673 a^4 b^20 c^16-1757 a^2 b^22 c^16-624 b^24 c^16+1716 a^22 c^18+869 a^20 b^2 c^18+8577 a^18 b^4 c^18+4971 a^16 b^6 c^18+949 a^14 b^8 c^18+1103 a^12 b^10 c^18+1337 a^10 b^12 c^18+1501 a^8 b^14 c^18+4512 a^6 b^16 c^18+5898 a^4 b^18 c^18-847 a^2 b^20 c^18+6006 b^22 c^18-4290 a^20 c^20-4312 a^18 b^2 c^20-9164 a^16 b^4 c^20-11454 a^14 b^6 c^20-10747 a^12 b^8 c^20-10625 a^10 b^10 c^20-10255 a^8 b^12 c^20-9339 a^6 b^14 c^20-4673 a^4 b^16 c^20-847 a^2 b^18 c^20-8580 b^20 c^20+4290 a^18 c^22+3949 a^16 b^2 c^22+3100 a^14 b^4 c^22+3138 a^12 b^6 c^22+2473 a^10 b^8 c^22+1339 a^8 b^10 c^22-843 a^6 b^12 c^22-3821 a^4 b^14 c^22-1757 a^2 b^16 c^22+6006 b^18 c^22-1716 a^16 c^24+2850 a^14 b^2 c^24+12006 a^12 b^4 c^24+19666 a^10 b^6 c^24+23737 a^8 b^8 c^24+23863 a^6 b^10 c^24+19803 a^4 b^12 c^24+10047 a^2 b^14 c^24-624 b^16 c^24-1092 a^14 c^26-9689 a^12 b^2 c^26-23586 a^10 b^4 c^26-34263 a^8 b^6 c^26-36453 a^6 b^8 c^26-29442 a^4 b^10 c^26-15911 a^2 b^12 c^26-3276 b^14 c^26+2184 a^12 c^28+10470 a^10 b^2 c^28+21507 a^8 b^4 c^28+27431 a^6 b^6 c^28+24201 a^4 b^8 c^28+14043 a^2 b^10 c^28+3864 b^12 c^28-1680 a^10 c^30-6529 a^8 b^2 c^30-11481 a^6 b^4 c^30-12099 a^4 b^6 c^30-7843 a^2 b^8 c^30-2460 b^10 c^30+780 a^8 c^32+2538 a^6 b^2 c^32+3638 a^4 b^4 c^32+2820 a^2 b^6 c^32+1008 b^8 c^32-228 a^6 c^34-595 a^4 b^2 c^34-622 a^2 b^4 c^34-267 b^6 c^34+39 a^4 c^36+73 a^2 b^2 c^36+42 b^4 c^36-3 a^2 c^38-3 b^2 c^38::
on line {2,3}
Searches = {-0.301845848082404806578171000150,-1.17056949401128116721702942740,4.59037221533791443730468593692}
Best regards,
Peter Moses.
Denote:
Na, Oa = the midpoints of NA, OA, resp.
N1, O1 = the reflections of Na, Oa in OA, NA, resp.
A' = NaOa /\ N1O1
Nb, Ob = the midpoints of NB, OB, resp.
N2, O2 = the reflections of Nb, Ob in OB, NB, resp.
B' = NbOb /\ N2O2
Nc, Oc = the midpoints of NC, OC, resp.
N3, O3 = the reflections of Nc, Oc in OC, NC, resp.
C' = NcOc /\ N3O3
The centroid of A'B'C' lies on the Euler line of ABC.
[Peter Moses]:
B' = NbOb /\ N2O2
Nc, Oc = the midpoints of NC, OC, resp.
N3, O3 = the reflections of Nc, Oc in OC, NC, resp.
C' = NcOc /\ N3O3
The centroid of A'B'C' lies on the Euler line of ABC.
[Peter Moses]:
Hi Antreas,
A very long-winded point on Euler line:
3 a^36 b^4-39 a^34 b^6+228 a^32 b^8-780 a^30 b^10+1680 a^28 b^12-2184 a^26 b^14+1092 a^24 b^16+1716 a^22 b^18-4290 a^20 b^20+4290 a^18 b^22-1716 a^16 b^24-1092 a^14 b^26+2184 a^12 b^28-1680 a^10 b^30+780 a^8 b^32-228 a^6 b^34+39 a^4 b^36-3 a^2 b^38+10 a^36 b^2 c^2-145 a^34 b^4 c^2+882 a^32 b^6 c^2-2986 a^30 b^8 c^2+6141 a^28 b^10 c^2-7622 a^26 b^12 c^2+4965 a^24 b^14 c^2-866 a^22 b^16 c^2+869 a^20 b^18 c^2-4312 a^18 b^20 c^2+3949 a^16 b^22 c^2+2850 a^14 b^24 c^2-9689 a^12 b^26 c^2+10470 a^10 b^28 c^2-6529 a^8 b^30 c^2+2538 a^6 b^32 c^2-595 a^4 b^34 c^2+73 a^2 b^36 c^2-3 b^38 c^2+3 a^36 c^4-145 a^34 b^2 c^4+1316 a^32 b^4 c^4-5418 a^30 b^6 c^4+12216 a^28 b^8 c^4-15408 a^26 b^10 c^4+8907 a^24 b^12 c^4+1975 a^22 b^14 c^4-7463 a^20 b^16 c^4+8577 a^18 b^18 c^4-9164 a^16 b^20 c^4+3100 a^14 b^22 c^4+12006 a^12 b^24 c^4-23586 a^10 b^26 c^4+21507 a^8 b^28 c^4-11481 a^6 b^30 c^4+3638 a^4 b^32 c^4-622 a^2 b^34 c^4+42 b^36 c^4-39 a^34 c^6+882 a^32 b^2 c^6-5418 a^30 b^4 c^6+15158 a^28 b^6 c^6-21282 a^26 b^8 c^6+12037 a^24 b^10 c^6+4905 a^22 b^12 c^6-11061 a^20 b^14 c^6+4875 a^18 b^16 c^6+4971 a^16 b^18 c^6-11454 a^14 b^20 c^6+3138 a^12 b^22 c^6+19666 a^10 b^24 c^6-34263 a^8 b^26 c^6+27431 a^6 b^28 c^6-12099 a^4 b^30 c^6+2820 a^2 b^32 c^6-267 b^34 c^6+228 a^32 c^8-2986 a^30 b^2 c^8+12216 a^28 b^4 c^8-21282 a^26 b^6 c^8+13390 a^24 b^8 c^8+5662 a^22 b^10 c^8-10450 a^20 b^12 c^8+472 a^18 b^14 c^8+5425 a^16 b^16 c^8+949 a^14 b^18 c^8-10747 a^12 b^20 c^8+2473 a^10 b^22 c^8+23737 a^8 b^24 c^8-36453 a^6 b^26 c^8+24201 a^4 b^28 c^8-7843 a^2 b^30 c^8+1008 b^32 c^8-780 a^30 c^10+6141 a^28 b^2 c^10-15408 a^26 b^4 c^10+12037 a^24 b^6 c^10+5662 a^22 b^8 c^10-10298 a^20 b^10 c^10+266 a^18 b^12 c^10+2380 a^16 b^14 c^10+2179 a^14 b^16 c^10+1103 a^12 b^18 c^10-10625 a^10 b^20 c^10+1339 a^8 b^22 c^10+23863 a^6 b^24 c^10-29442 a^4 b^26 c^10+14043 a^2 b^28 c^10-2460 b^30 c^10+1680 a^28 c^12-7622 a^26 b^2 c^12+8907 a^24 b^4 c^12+4905 a^22 b^6 c^12-10450 a^20 b^8 c^12+266 a^18 b^10 c^12+2758 a^16 b^12 c^12-996 a^14 b^14 c^12+2557 a^12 b^16 c^12+1337 a^10 b^18 c^12-10255 a^8 b^20 c^12-843 a^6 b^22 c^12+19803 a^4 b^24 c^12-15911 a^2 b^26 c^12+3864 b^28 c^12-2184 a^26 c^14+4965 a^24 b^2 c^14+1975 a^22 b^4 c^14-11061 a^20 b^6 c^14+472 a^18 b^8 c^14+2380 a^16 b^10 c^14-996 a^14 b^12 c^14-1104 a^12 b^14 c^14+1945 a^10 b^16 c^14+1501 a^8 b^18 c^14-9339 a^6 b^20 c^14-3821 a^4 b^22 c^14+10047 a^2 b^24 c^14-3276 b^26 c^14+1092 a^24 c^16-866 a^22 b^2 c^16-7463 a^20 b^4 c^16+4875 a^18 b^6 c^16+5425 a^16 b^8 c^16+2179 a^14 b^10 c^16+2557 a^12 b^12 c^16+1945 a^10 b^14 c^16+4366 a^8 b^16 c^16+4512 a^6 b^18 c^16-4673 a^4 b^20 c^16-1757 a^2 b^22 c^16-624 b^24 c^16+1716 a^22 c^18+869 a^20 b^2 c^18+8577 a^18 b^4 c^18+4971 a^16 b^6 c^18+949 a^14 b^8 c^18+1103 a^12 b^10 c^18+1337 a^10 b^12 c^18+1501 a^8 b^14 c^18+4512 a^6 b^16 c^18+5898 a^4 b^18 c^18-847 a^2 b^20 c^18+6006 b^22 c^18-4290 a^20 c^20-4312 a^18 b^2 c^20-9164 a^16 b^4 c^20-11454 a^14 b^6 c^20-10747 a^12 b^8 c^20-10625 a^10 b^10 c^20-10255 a^8 b^12 c^20-9339 a^6 b^14 c^20-4673 a^4 b^16 c^20-847 a^2 b^18 c^20-8580 b^20 c^20+4290 a^18 c^22+3949 a^16 b^2 c^22+3100 a^14 b^4 c^22+3138 a^12 b^6 c^22+2473 a^10 b^8 c^22+1339 a^8 b^10 c^22-843 a^6 b^12 c^22-3821 a^4 b^14 c^22-1757 a^2 b^16 c^22+6006 b^18 c^22-1716 a^16 c^24+2850 a^14 b^2 c^24+12006 a^12 b^4 c^24+19666 a^10 b^6 c^24+23737 a^8 b^8 c^24+23863 a^6 b^10 c^24+19803 a^4 b^12 c^24+10047 a^2 b^14 c^24-624 b^16 c^24-1092 a^14 c^26-9689 a^12 b^2 c^26-23586 a^10 b^4 c^26-34263 a^8 b^6 c^26-36453 a^6 b^8 c^26-29442 a^4 b^10 c^26-15911 a^2 b^12 c^26-3276 b^14 c^26+2184 a^12 c^28+10470 a^10 b^2 c^28+21507 a^8 b^4 c^28+27431 a^6 b^6 c^28+24201 a^4 b^8 c^28+14043 a^2 b^10 c^28+3864 b^12 c^28-1680 a^10 c^30-6529 a^8 b^2 c^30-11481 a^6 b^4 c^30-12099 a^4 b^6 c^30-7843 a^2 b^8 c^30-2460 b^10 c^30+780 a^8 c^32+2538 a^6 b^2 c^32+3638 a^4 b^4 c^32+2820 a^2 b^6 c^32+1008 b^8 c^32-228 a^6 c^34-595 a^4 b^2 c^34-622 a^2 b^4 c^34-267 b^6 c^34+39 a^4 c^36+73 a^2 b^2 c^36+42 b^4 c^36-3 a^2 c^38-3 b^2 c^38::
on line {2,3}
Searches = {-0.301845848082404806578171000150,-1.17056949401128116721702942740,4.59037221533791443730468593692}
Best regards,
Peter Moses.
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