[Antreas P. Hatzipolakis]:
Let ABC be a triangle.
Denote:
A'B'C' = the pedal triangle of O
A"B"C" = the pedal triangle of O' [=circumcenter of A'B'C' = N] wrt triangle A'B'C'
(ie A'B'C' = the medial tiangle of ABC, A"B"C"= the medial triangle of A'B'C')
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
The triangles A"B"C", NaNbNc are circumcyclologic.
1. Cyclologic centers?
Denote:
R1 = the radical axis of the circumcircles of (NbC"A"), (NcA"B")
Denote:
A'B'C' = the pedal triangle of O
A"B"C" = the pedal triangle of O' [=circumcenter of A'B'C' = N] wrt triangle A'B'C'
(ie A'B'C' = the medial tiangle of ABC, A"B"C"= the medial triangle of A'B'C')
Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.
The triangles A"B"C", NaNbNc are circumcyclologic.
ie the circumcircles of A"B"C", A"NbNc, B"NcNa, C"NaNb are concurrent.
the circumcircles of NaNbNc, NaB"C", NbC"A", NcA"B" are concurrent.
the circumcircles of NaNbNc, NaB"C", NbC"A", NcA"B" are concurrent.
1. Cyclologic centers?
Denote:
R1 = the radical axis of the circumcircles of (NbC"A"), (NcA"B")
R2 = the radical axis of the circumcircles of (NcA"B"), (NaB"C")
R3 = the radical axis of the circumcircles of (NaB"C"), (NbC"A")
2. The parallels to R1, R2, R3 through A, B, C, resp. are concurrent
3. The parallels to R1, R2, R3 through A', B', C', resp. are concurrent
[Peter Moses]:
Hi Antreas,
1).
A"NbNc.
(a^2-b^2-c^2) (2 a^22-12 a^20 b^2+27 a^18 b^4-21 a^16 b^6-20 a^14 b^8+56 a^12 b^10-42 a^10 b^12-2 a^8 b^14+26 a^6 b^16-20 a^4 b^18+7 a^2 b^20-b^22-12 a^20 c^2+60 a^18 b^2 c^2-117 a^16 b^4 c^2+112 a^14 b^6 c^2-52 a^12 b^8 c^2-12 a^10 b^10 c^2+78 a^8 b^12 c^2-120 a^6 b^14 c^2+96 a^4 b^16 c^2-40 a^2 b^18 c^2+7 b^20 c^2+27 a^18 c^4-117 a^16 b^2 c^4+184 a^14 b^4 c^4-140 a^12 b^6 c^4+82 a^10 b^8 c^4-114 a^8 b^10 c^4+188 a^6 b^12 c^4-184 a^4 b^14 c^4+95 a^2 b^16 c^4-21 b^18 c^4-21 a^16 c^6+112 a^14 b^2 c^6-140 a^12 b^4 c^6+48 a^10 b^6 c^6+30 a^8 b^8 c^6-136 a^6 b^10 c^6+176 a^4 b^12 c^6-120 a^2 b^14 c^6+35 b^16 c^6-20 a^14 c^8-52 a^12 b^2 c^8+82 a^10 b^4 c^8+30 a^8 b^6 c^8+84 a^6 b^8 c^8-68 a^4 b^10 c^8+90 a^2 b^12 c^8-34 b^14 c^8+56 a^12 c^10-12 a^10 b^2 c^10-114 a^8 b^4 c^10-136 a^6 b^6 c^10-68 a^4 b^8 c^10-64 a^2 b^10 c^10+14 b^12 c^10-42 a^10 c^12+78 a^8 b^2 c^12+188 a^6 b^4 c^12+176 a^4 b^6 c^12+90 a^2 b^8 c^12+14 b^10 c^12-2 a^8 c^14-120 a^6 b^2 c^14-184 a^4 b^4 c^14-120 a^2 b^6 c^14-34 b^8 c^14+26 a^6 c^16+96 a^4 b^2 c^16+95 a^2 b^4 c^16+35 b^6 c^16-20 a^4 c^18-40 a^2 b^2 c^18-21 b^4 c^18+7 a^2 c^20+7 b^2 c^20-c^22)::
Hi Antreas,
1).
A"NbNc.
(a^2-b^2-c^2) (2 a^22-12 a^20 b^2+27 a^18 b^4-21 a^16 b^6-20 a^14 b^8+56 a^12 b^10-42 a^10 b^12-2 a^8 b^14+26 a^6 b^16-20 a^4 b^18+7 a^2 b^20-b^22-12 a^20 c^2+60 a^18 b^2 c^2-117 a^16 b^4 c^2+112 a^14 b^6 c^2-52 a^12 b^8 c^2-12 a^10 b^10 c^2+78 a^8 b^12 c^2-120 a^6 b^14 c^2+96 a^4 b^16 c^2-40 a^2 b^18 c^2+7 b^20 c^2+27 a^18 c^4-117 a^16 b^2 c^4+184 a^14 b^4 c^4-140 a^12 b^6 c^4+82 a^10 b^8 c^4-114 a^8 b^10 c^4+188 a^6 b^12 c^4-184 a^4 b^14 c^4+95 a^2 b^16 c^4-21 b^18 c^4-21 a^16 c^6+112 a^14 b^2 c^6-140 a^12 b^4 c^6+48 a^10 b^6 c^6+30 a^8 b^8 c^6-136 a^6 b^10 c^6+176 a^4 b^12 c^6-120 a^2 b^14 c^6+35 b^16 c^6-20 a^14 c^8-52 a^12 b^2 c^8+82 a^10 b^4 c^8+30 a^8 b^6 c^8+84 a^6 b^8 c^8-68 a^4 b^10 c^8+90 a^2 b^12 c^8-34 b^14 c^8+56 a^12 c^10-12 a^10 b^2 c^10-114 a^8 b^4 c^10-136 a^6 b^6 c^10-68 a^4 b^8 c^10-64 a^2 b^10 c^10+14 b^12 c^10-42 a^10 c^12+78 a^8 b^2 c^12+188 a^6 b^4 c^12+176 a^4 b^6 c^12+90 a^2 b^8 c^12+14 b^10 c^12-2 a^8 c^14-120 a^6 b^2 c^14-184 a^4 b^4 c^14-120 a^2 b^6 c^14-34 b^8 c^14+26 a^6 c^16+96 a^4 b^2 c^16+95 a^2 b^4 c^16+35 b^6 c^16-20 a^4 c^18-40 a^2 b^2 c^18-21 b^4 c^18+7 a^2 c^20+7 b^2 c^20-c^22)::
on line {6699,10116}.
Searches: {4.85785120633709213951659419558,2.11555071196226464581450185470,-0.0660327216066158699147273487116}.
NaB"C".
a^24 b^4-10 a^22 b^6+44 a^20 b^8-110 a^18 b^10+165 a^16 b^12-132 a^14 b^14+132 a^10 b^18-165 a^8 b^20+110 a^6 b^22-44 a^4 b^24+10 a^2 b^26-b^28-4 a^24 b^2 c^2+24 a^22 b^4 c^2-54 a^20 b^6 c^2+44 a^18 b^8 c^2+42 a^16 b^10 c^2-192 a^14 b^12 c^2+420 a^12 b^14 c^2-696 a^10 b^16 c^2+816 a^8 b^18 c^2-632 a^6 b^20 c^2+306 a^4 b^22 c^2-84 a^2 b^24 c^2+10 b^26 c^2+a^24 c^4+24 a^22 b^2 c^4-140 a^20 b^4 c^4+306 a^18 b^6 c^4-357 a^16 b^8 c^4+308 a^14 b^10 c^4-416 a^12 b^12 c^4+848 a^10 b^14 c^4-1361 a^8 b^16 c^4+1412 a^6 b^18 c^4-884 a^4 b^20 c^4+302 a^2 b^22 c^4-43 b^24 c^4-10 a^22 c^6-54 a^20 b^2 c^6+306 a^18 b^4 c^6-508 a^16 b^6 c^6+392 a^14 b^8 c^6-108 a^12 b^10 c^6-296 a^10 b^12 c^6+984 a^8 b^14 c^6-1582 a^6 b^16 c^6+1378 a^4 b^18 c^6-602 a^2 b^20 c^6+100 b^22 c^6+44 a^20 c^8+44 a^18 b^2 c^8-357 a^16 b^4 c^8+392 a^14 b^6 c^8-144 a^12 b^8 c^8+44 a^10 b^10 c^8-298 a^8 b^12 c^8+944 a^6 b^14 c^8-1268 a^4 b^16 c^8+720 a^2 b^18 c^8-121 b^20 c^8-110 a^18 c^10+42 a^16 b^2 c^10+308 a^14 b^4 c^10-108 a^12 b^6 c^10+44 a^10 b^8 c^10+48 a^8 b^10 c^10-252 a^6 b^12 c^10+748 a^4 b^14 c^10-518 a^2 b^16 c^10+22 b^18 c^10+165 a^16 c^12-192 a^14 b^2 c^12-416 a^12 b^4 c^12-296 a^10 b^6 c^12-298 a^8 b^8 c^12-252 a^6 b^10 c^12-472 a^4 b^12 c^12+172 a^2 b^14 c^12+165 b^16 c^12-132 a^14 c^14+420 a^12 b^2 c^14+848 a^10 b^4 c^14+984 a^8 b^6 c^14+944 a^6 b^8 c^14+748 a^4 b^10 c^14+172 a^2 b^12 c^14-264 b^14 c^14-696 a^10 b^2 c^16-1361 a^8 b^4 c^16-1582 a^6 b^6 c^16-1268 a^4 b^8 c^16-518 a^2 b^10 c^16+165 b^12 c^16+132 a^10 c^18+816 a^8 b^2 c^18+1412 a^6 b^4 c^18+1378 a^4 b^6 c^18+720 a^2 b^8 c^18+22 b^10 c^18-165 a^8 c^20-632 a^6 b^2 c^20-884 a^4 b^4 c^20-602 a^2 b^6 c^20-121 b^8 c^20+110 a^6 c^22+306 a^4 b^2 c^22+302 a^2 b^4 c^22+100 b^6 c^22-44 a^4 c^24-84 a^2 b^2 c^24-43 b^4 c^24+10 a^2 c^26+10 b^2 c^26-c^28::
Searches: {1.97758646397124305413159842541,2.77995375286296578117901323603,0.803348900400213691164455127367}.
2)
(a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) (a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6)::
Searches: {4.85785120633709213951659419558,2.11555071196226464581450185470,-0.0660327216066158699147273487116}.
NaB"C".
a^24 b^4-10 a^22 b^6+44 a^20 b^8-110 a^18 b^10+165 a^16 b^12-132 a^14 b^14+132 a^10 b^18-165 a^8 b^20+110 a^6 b^22-44 a^4 b^24+10 a^2 b^26-b^28-4 a^24 b^2 c^2+24 a^22 b^4 c^2-54 a^20 b^6 c^2+44 a^18 b^8 c^2+42 a^16 b^10 c^2-192 a^14 b^12 c^2+420 a^12 b^14 c^2-696 a^10 b^16 c^2+816 a^8 b^18 c^2-632 a^6 b^20 c^2+306 a^4 b^22 c^2-84 a^2 b^24 c^2+10 b^26 c^2+a^24 c^4+24 a^22 b^2 c^4-140 a^20 b^4 c^4+306 a^18 b^6 c^4-357 a^16 b^8 c^4+308 a^14 b^10 c^4-416 a^12 b^12 c^4+848 a^10 b^14 c^4-1361 a^8 b^16 c^4+1412 a^6 b^18 c^4-884 a^4 b^20 c^4+302 a^2 b^22 c^4-43 b^24 c^4-10 a^22 c^6-54 a^20 b^2 c^6+306 a^18 b^4 c^6-508 a^16 b^6 c^6+392 a^14 b^8 c^6-108 a^12 b^10 c^6-296 a^10 b^12 c^6+984 a^8 b^14 c^6-1582 a^6 b^16 c^6+1378 a^4 b^18 c^6-602 a^2 b^20 c^6+100 b^22 c^6+44 a^20 c^8+44 a^18 b^2 c^8-357 a^16 b^4 c^8+392 a^14 b^6 c^8-144 a^12 b^8 c^8+44 a^10 b^10 c^8-298 a^8 b^12 c^8+944 a^6 b^14 c^8-1268 a^4 b^16 c^8+720 a^2 b^18 c^8-121 b^20 c^8-110 a^18 c^10+42 a^16 b^2 c^10+308 a^14 b^4 c^10-108 a^12 b^6 c^10+44 a^10 b^8 c^10+48 a^8 b^10 c^10-252 a^6 b^12 c^10+748 a^4 b^14 c^10-518 a^2 b^16 c^10+22 b^18 c^10+165 a^16 c^12-192 a^14 b^2 c^12-416 a^12 b^4 c^12-296 a^10 b^6 c^12-298 a^8 b^8 c^12-252 a^6 b^10 c^12-472 a^4 b^12 c^12+172 a^2 b^14 c^12+165 b^16 c^12-132 a^14 c^14+420 a^12 b^2 c^14+848 a^10 b^4 c^14+984 a^8 b^6 c^14+944 a^6 b^8 c^14+748 a^4 b^10 c^14+172 a^2 b^12 c^14-264 b^14 c^14-696 a^10 b^2 c^16-1361 a^8 b^4 c^16-1582 a^6 b^6 c^16-1268 a^4 b^8 c^16-518 a^2 b^10 c^16+165 b^12 c^16+132 a^10 c^18+816 a^8 b^2 c^18+1412 a^6 b^4 c^18+1378 a^4 b^6 c^18+720 a^2 b^8 c^18+22 b^10 c^18-165 a^8 c^20-632 a^6 b^2 c^20-884 a^4 b^4 c^20-602 a^2 b^6 c^20-121 b^8 c^20+110 a^6 c^22+306 a^4 b^2 c^22+302 a^2 b^4 c^22+100 b^6 c^22-44 a^4 c^24-84 a^2 b^2 c^24-43 b^4 c^24+10 a^2 c^26+10 b^2 c^26-c^28::
Searches: {1.97758646397124305413159842541,2.77995375286296578117901323603,0.803348900400213691164455127367}.
2)
(a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) (a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6)::
on lines {{68,2072},{5962,13579}}.
X(921)-isoconjugate of X(2904).
barycentric quotient X(1609)/X(2904).
Searches: {0.0222394784188174931032893129074,5.86107742647442664243398335160,-0.427268880306601261537458439184}.
3)
a^2 (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^16-5 a^14 b^2+9 a^12 b^4-5 a^10 b^6-5 a^8 b^8+9 a^6 b^10-5 a^4 b^12+a^2 b^14-5 a^14 c^2+20 a^12 b^2 c^2-29 a^10 b^4 c^2+22 a^8 b^6 c^2-15 a^6 b^8 c^2+8 a^4 b^10 c^2+a^2 b^12 c^2-2 b^14 c^2+9 a^12 c^4-29 a^10 b^2 c^4+22 a^8 b^4 c^4-2 a^6 b^6 c^4-3 a^4 b^8 c^4-9 a^2 b^10 c^4+12 b^12 c^4-5 a^10 c^6+22 a^8 b^2 c^6-2 a^6 b^4 c^6+7 a^2 b^8 c^6-30 b^10 c^6-5 a^8 c^8-15 a^6 b^2 c^8-3 a^4 b^4 c^8+7 a^2 b^6 c^8+40 b^8 c^8+9 a^6 c^10+8 a^4 b^2 c^10-9 a^2 b^4 c^10-30 b^6 c^10-5 a^4 c^12+a^2 b^2 c^12+12 b^4 c^12+a^2 c^14-2 b^2 c^14)::
Searches: {3.93293344952366861515990753792,-0.301169920748495080075956879553,2.03396668110702864386636869392}.
Best regards,
Peter Moses.
X(921)-isoconjugate of X(2904).
barycentric quotient X(1609)/X(2904).
Searches: {0.0222394784188174931032893129074,5.86107742647442664243398335160,-0.427268880306601261537458439184}.
3)
a^2 (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) (a^16-5 a^14 b^2+9 a^12 b^4-5 a^10 b^6-5 a^8 b^8+9 a^6 b^10-5 a^4 b^12+a^2 b^14-5 a^14 c^2+20 a^12 b^2 c^2-29 a^10 b^4 c^2+22 a^8 b^6 c^2-15 a^6 b^8 c^2+8 a^4 b^10 c^2+a^2 b^12 c^2-2 b^14 c^2+9 a^12 c^4-29 a^10 b^2 c^4+22 a^8 b^4 c^4-2 a^6 b^6 c^4-3 a^4 b^8 c^4-9 a^2 b^10 c^4+12 b^12 c^4-5 a^10 c^6+22 a^8 b^2 c^6-2 a^6 b^4 c^6+7 a^2 b^8 c^6-30 b^10 c^6-5 a^8 c^8-15 a^6 b^2 c^8-3 a^4 b^4 c^8+7 a^2 b^6 c^8+40 b^8 c^8+9 a^6 c^10+8 a^4 b^2 c^10-9 a^2 b^4 c^10-30 b^6 c^10-5 a^4 c^12+a^2 b^2 c^12+12 b^4 c^12+a^2 c^14-2 b^2 c^14)::
Searches: {3.93293344952366861515990753792,-0.301169920748495080075956879553,2.03396668110702864386636869392}.
Best regards,
Peter Moses.
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