[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C', A"B"C" the antimedial, medial triangle, resp.
Denote:
Na, Nb, Nc = the NPC centers of HB'C', HC'A', HA'B', resp. (H of ABC = O of A'B'C')
Denote:
Na, Nb, Nc = the NPC centers of HB'C', HC'A', HA'B', resp. (H of ABC = O of A'B'C')
The triangles A"B"C", NaNbNc are circumcyclologic.
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]:
(b-c)^2 (b+c)^2 (-a^2+b^2-c^2) (a^2+b^2-c^2) (-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^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+6 a^6 b^2 c^2-6 a^4 b^4 c^2+6 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-6 a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4+2 a^4 c^6+6 a^2 b^2 c^6+2 b^4 c^6-3 a^2 c^8-3 b^2 c^8+c^10)::
Searches: {-1.82759603520802955561008400241,1.02763616105290719065306588311,3.77272992512068776602473364608}.
NaB"C"
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}.
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: {0.0222394784188174931032893129074,5.86107742647442664243398335160,-0.427268880306601261537458439184}.
3)
3 a^28-30 a^26 b^2+133 a^24 b^4-340 a^22 b^6+539 a^20 b^8-506 a^18 b^10+165 a^16 b^12+264 a^14 b^14-495 a^12 b^16+462 a^10 b^18-297 a^8 b^20+140 a^6 b^22-47 a^4 b^24+10 a^2 b^26-b^28-30 a^26 c^2+242 a^24 b^2 c^2-852 a^22 b^4 c^2+1716 a^20 b^6 c^2-2194 a^18 b^8 c^2+1926 a^16 b^10 c^2-1368 a^14 b^12 c^2+1080 a^12 b^14 c^2-1026 a^10 b^16 c^2+894 a^8 b^18 c^2-596 a^6 b^20 c^2+276 a^4 b^22 c^2-78 a^2 b^24 c^2+10 b^26 c^2+133 a^24 c^4-852 a^22 b^2 c^4+2302 a^20 b^4 c^4-3420 a^18 b^6 c^4+3051 a^16 b^8 c^4-1624 a^14 b^10 c^4+292 a^12 b^12 c^4+536 a^10 b^14 c^4-965 a^8 b^16 c^4+1004 a^6 b^18 c^4-674 a^4 b^20 c^4+260 a^2 b^22 c^4-43 b^24 c^4-340 a^22 c^6+1716 a^20 b^2 c^6-3420 a^18 b^4 c^6+3476 a^16 b^6 c^6-1912 a^14 b^8 c^6+552 a^12 b^10 c^6-200 a^10 b^12 c^6+456 a^8 b^14 c^6-820 a^6 b^16 c^6+868 a^4 b^18 c^6-476 a^2 b^20 c^6+100 b^22 c^6+539 a^20 c^8-2194 a^18 b^2 c^8+3051 a^16 b^4 c^8-1912 a^14 b^6 c^8+534 a^12 b^8 c^8-28 a^10 b^10 c^8-82 a^8 b^12 c^8+296 a^6 b^14 c^8-593 a^4 b^16 c^8+510 a^2 b^18 c^8-121 b^20 c^8-506 a^18 c^10+1926 a^16 b^2 c^10-1624 a^14 b^4 c^10+552 a^12 b^6 c^10-28 a^10 b^8 c^10-12 a^8 b^10 c^10-24 a^6 b^12 c^10+136 a^4 b^14 c^10-314 a^2 b^16 c^10+22 b^18 c^10+165 a^16 c^12-1368 a^14 b^2 c^12+292 a^12 b^4 c^12-200 a^10 b^6 c^12-82 a^8 b^8 c^12-24 a^6 b^10 c^12+68 a^4 b^12 c^12+88 a^2 b^14 c^12+165 b^16 c^12+264 a^14 c^14+1080 a^12 b^2 c^14+536 a^10 b^4 c^14+456 a^8 b^6 c^14+296 a^6 b^8 c^14+136 a^4 b^10 c^14+88 a^2 b^12 c^14-264 b^14 c^14-495 a^12 c^16-1026 a^10 b^2 c^16-965 a^8 b^4 c^16-820 a^6 b^6 c^16-593 a^4 b^8 c^16-314 a^2 b^10 c^16+165 b^12 c^16+462 a^10 c^18+894 a^8 b^2 c^18+1004 a^6 b^4 c^18+868 a^4 b^6 c^18+510 a^2 b^8 c^18+22 b^10 c^18-297 a^8 c^20-596 a^6 b^2 c^20-674 a^4 b^4 c^20-476 a^2 b^6 c^20-121 b^8 c^20+140 a^6 c^22+276 a^4 b^2 c^22+260 a^2 b^4 c^22+100 b^6 c^22-47 a^4 c^24-78 a^2 b^2 c^24-43 b^4 c^24+10 a^2 c^26+10 b^2 c^26-c^28::
Searches: {7.84362742062851973721652576293,-6.46341726797141680258589711071,4.49520224252065854927019582702}.
ie the circumcircles of A"B"C"[ = NPC of ABC] , 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"NbNa:
1)
A"NbNa:
(b-c)^2 (b+c)^2 (-a^2+b^2-c^2) (a^2+b^2-c^2) (-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^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+6 a^6 b^2 c^2-6 a^4 b^4 c^2+6 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-6 a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4+2 a^4 c^6+6 a^2 b^2 c^6+2 b^4 c^6-3 a^2 c^8-3 b^2 c^8+c^10)::
on the 9-point-circle and the line {113,2904}..
Searches: {-1.82759603520802955561008400241,1.02763616105290719065306588311,3.77272992512068776602473364608}.
NaB"C"
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}.
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).
X(921)-isoconjugate of X(2904).
barycentric quotient X(1609)/X(2904).
Searches: {0.0222394784188174931032893129074,5.86107742647442664243398335160,-0.427268880306601261537458439184}.
3)
3 a^28-30 a^26 b^2+133 a^24 b^4-340 a^22 b^6+539 a^20 b^8-506 a^18 b^10+165 a^16 b^12+264 a^14 b^14-495 a^12 b^16+462 a^10 b^18-297 a^8 b^20+140 a^6 b^22-47 a^4 b^24+10 a^2 b^26-b^28-30 a^26 c^2+242 a^24 b^2 c^2-852 a^22 b^4 c^2+1716 a^20 b^6 c^2-2194 a^18 b^8 c^2+1926 a^16 b^10 c^2-1368 a^14 b^12 c^2+1080 a^12 b^14 c^2-1026 a^10 b^16 c^2+894 a^8 b^18 c^2-596 a^6 b^20 c^2+276 a^4 b^22 c^2-78 a^2 b^24 c^2+10 b^26 c^2+133 a^24 c^4-852 a^22 b^2 c^4+2302 a^20 b^4 c^4-3420 a^18 b^6 c^4+3051 a^16 b^8 c^4-1624 a^14 b^10 c^4+292 a^12 b^12 c^4+536 a^10 b^14 c^4-965 a^8 b^16 c^4+1004 a^6 b^18 c^4-674 a^4 b^20 c^4+260 a^2 b^22 c^4-43 b^24 c^4-340 a^22 c^6+1716 a^20 b^2 c^6-3420 a^18 b^4 c^6+3476 a^16 b^6 c^6-1912 a^14 b^8 c^6+552 a^12 b^10 c^6-200 a^10 b^12 c^6+456 a^8 b^14 c^6-820 a^6 b^16 c^6+868 a^4 b^18 c^6-476 a^2 b^20 c^6+100 b^22 c^6+539 a^20 c^8-2194 a^18 b^2 c^8+3051 a^16 b^4 c^8-1912 a^14 b^6 c^8+534 a^12 b^8 c^8-28 a^10 b^10 c^8-82 a^8 b^12 c^8+296 a^6 b^14 c^8-593 a^4 b^16 c^8+510 a^2 b^18 c^8-121 b^20 c^8-506 a^18 c^10+1926 a^16 b^2 c^10-1624 a^14 b^4 c^10+552 a^12 b^6 c^10-28 a^10 b^8 c^10-12 a^8 b^10 c^10-24 a^6 b^12 c^10+136 a^4 b^14 c^10-314 a^2 b^16 c^10+22 b^18 c^10+165 a^16 c^12-1368 a^14 b^2 c^12+292 a^12 b^4 c^12-200 a^10 b^6 c^12-82 a^8 b^8 c^12-24 a^6 b^10 c^12+68 a^4 b^12 c^12+88 a^2 b^14 c^12+165 b^16 c^12+264 a^14 c^14+1080 a^12 b^2 c^14+536 a^10 b^4 c^14+456 a^8 b^6 c^14+296 a^6 b^8 c^14+136 a^4 b^10 c^14+88 a^2 b^12 c^14-264 b^14 c^14-495 a^12 c^16-1026 a^10 b^2 c^16-965 a^8 b^4 c^16-820 a^6 b^6 c^16-593 a^4 b^8 c^16-314 a^2 b^10 c^16+165 b^12 c^16+462 a^10 c^18+894 a^8 b^2 c^18+1004 a^6 b^4 c^18+868 a^4 b^6 c^18+510 a^2 b^8 c^18+22 b^10 c^18-297 a^8 c^20-596 a^6 b^2 c^20-674 a^4 b^4 c^20-476 a^2 b^6 c^20-121 b^8 c^20+140 a^6 c^22+276 a^4 b^2 c^22+260 a^2 b^4 c^22+100 b^6 c^22-47 a^4 c^24-78 a^2 b^2 c^24-43 b^4 c^24+10 a^2 c^26+10 b^2 c^26-c^28::
on line {186,6193}.
Searches: {7.84362742062851973721652576293,-6.46341726797141680258589711071,4.49520224252065854927019582702}.
Best regards,
Peter Moses.
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