HYACINTHOS 29174

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of N.

 

Denote:

 

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

 

N1, N2, N3 = the orthogonal projections of Na, Nb, Nc on BC, CA, AB, resp.

 

MaMbMc = the medial triangle of NaNbNc

 

M1, M2, M3 = the reflections of Ma, Mb, Mc in NaN1, NbN2, NcN3, resp.

 

A'B'C', M1M2M3 are orthologic.

 

Orhologic centers?

 

 

[Ercole Suppa]:

 

 

Hi Antreas

 

Z(A'B'C', M1M2M3) =

 

= a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^14-7 a^12 b^2+21 a^10 b^4-35 a^8 b^6+35 a^6 b^8-21 a^4 b^10+7 a^2 b^12-b^14-7 a^12 c^2+24 a^10 b^2 c^2-29 a^8 b^4 c^2+8 a^6 b^6 c^2+15 a^4 b^8 c^2-16 a^2 b^10 c^2+5 b^12 c^2+21 a^10 c^4-29 a^8 b^2 c^4+7 a^6 b^4 c^4-3 a^4 b^6 c^4+13 a^2 b^8 c^4-9 b^10 c^4-35 a^8 c^6+8 a^6 b^2 c^6-3 a^4 b^4 c^6-8 a^2 b^6 c^6+5 b^8 c^6+35 a^6 c^8+15 a^4 b^2 c^8+13 a^2 b^4 c^8+5 b^6 c^8-21 a^4 c^10-16 a^2 b^2 c^10-9 b^4 c^10+7 a^2 c^12+5 b^2 c^12-c^14) (a^20-8 a^18 b^2+27 a^16 b^4-48 a^14 b^6+42 a^12 b^8-42 a^8 b^12+48 a^6 b^14-27 a^4 b^16+8 a^2 b^18-b^20-8 a^18 c^2+41 a^16 b^2 c^2-80 a^14 b^4 c^2+66 a^12 b^6 c^2-14 a^10 b^8 c^2+18 a^8 b^10 c^2-76 a^6 b^12 c^2+90 a^4 b^14 c^2-46 a^2 b^16 c^2+9 b^18 c^2+27 a^16 c^4-80 a^14 b^2 c^4+71 a^12 b^4 c^4-10 a^10 b^6 c^4-4 a^8 b^8 c^4+30 a^6 b^10 c^4-107 a^4 b^12 c^4+108 a^2 b^14 c^4-35 b^16 c^4-48 a^14 c^6+66 a^12 b^2 c^6-10 a^10 b^4 c^6-7 a^8 b^6 c^6-2 a^6 b^8 c^6+49 a^4 b^10 c^6-128 a^2 b^12 c^6+80 b^14 c^6+42 a^12 c^8-14 a^10 b^2 c^8-4 a^8 b^4 c^8-2 a^6 b^6 c^8-10 a^4 b^8 c^8+58 a^2 b^10 c^8-124 b^12 c^8+18 a^8 b^2 c^10+30 a^6 b^4 c^10+49 a^4 b^6 c^10+58 a^2 b^8 c^10+142 b^10 c^10-42 a^8 c^12-76 a^6 b^2 c^12-107 a^4 b^4 c^12-128 a^2 b^6 c^12-124 b^8 c^12+48 a^6 c^14+90 a^4 b^2 c^14+108 a^2 b^4 c^14+80 b^6 c^14-27 a^4 c^16-46 a^2 b^2 c^16-35 b^4 c^16+8 a^2 c^18+9 b^2 c^18-c^20) : : (barys)

 

= (220 R^4+60 R^2 SB+60 R^2 SC-116 R^2 SW-16 SB SW-16 SC SW+16 SW^2)S^6 + (920 R^8+185 R^6 SB+185 R^6 SC-100 R^4 SB SC-991 R^6 SW-156 R^4 SB SW-156 R^4 SC SW+100 R^2 SB SC SW+332 R^4 SW^2+28 R^2 SB SW^2+28 R^2 SC SW^2-16 SB SC SW^2-36 R^2 SW^3)S^4 + (780 R^12+140 R^10 SB+140 R^10 SC-600 R^8 SB SC-1279 R^10 SW-199 R^8 SB SW-199 R^8 SC SW+875 R^6 SB SC SW+747 R^8 SW^2+89 R^6 SB SW^2+89 R^6 SC SW^2-396 R^4 SB SC SW^2-183 R^6 SW^3-12 R^4 SB SW^3-12 R^4 SC SW^3+52 R^2 SB SC SW^3+16 R^4 SW^4)S^2 + 900 R^12 SB SC+1725 R^10 SB SC SW-1219 R^8 SB SC SW^2+371 R^6 SB SC SW^3-40 R^4 SB SC SW^4 : : (barys)

 

=(6-9-13)-search numbers [0.0378260284388059758, -0.2359953006758014263, 3.7865876769420234864]

 

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Z( M1M2M3,A'B'C') = (name pending) 

 

= a^2 (a^24 b^2-10 a^22 b^4+44 a^20 b^6-110 a^18 b^8+165 a^16 b^10-132 a^14 b^12+132 a^10 b^16-165 a^8 b^18+110 a^6 b^20-44 a^4 b^22+10 a^2 b^24-b^26+a^24 c^2-18 a^22 b^2 c^2+101 a^20 b^4 c^2-279 a^18 b^6 c^2+425 a^16 b^8 c^2-338 a^14 b^10 c^2+84 a^12 b^12 c^2+20 a^10 b^14 c^2+109 a^8 b^16 c^2-220 a^6 b^18 c^2+167 a^4 b^20 c^2-61 a^2 b^22 c^2+9 b^24 c^2-10 a^22 c^4+101 a^20 b^2 c^4-356 a^18 b^4 c^4+581 a^16 b^6 c^4-404 a^14 b^8 c^4-23 a^12 b^10 c^4+166 a^10 b^12 c^4-69 a^8 b^14 c^4+114 a^6 b^16 c^4-206 a^4 b^18 c^4+138 a^2 b^20 c^4-32 b^22 c^4+44 a^20 c^6-279 a^18 b^2 c^6+581 a^16 b^4 c^6-430 a^14 b^6 c^6-34 a^12 b^8 c^6+99 a^10 b^10 c^6+86 a^8 b^12 c^6-91 a^6 b^14 c^6+103 a^4 b^16 c^6-135 a^2 b^18 c^6+56 b^20 c^6-110 a^18 c^8+425 a^16 b^2 c^8-404 a^14 b^4 c^8-34 a^12 b^6 c^8+84 a^10 b^8 c^8+39 a^8 b^10 c^8+78 a^6 b^12 c^8-83 a^4 b^14 c^8+50 a^2 b^16 c^8-45 b^18 c^8+165 a^16 c^10-338 a^14 b^2 c^10-23 a^12 b^4 c^10+99 a^10 b^6 c^10+39 a^8 b^8 c^10+18 a^6 b^10 c^10+63 a^4 b^12 c^10-28 a^2 b^14 c^10+5 b^16 c^10-132 a^14 c^12+84 a^12 b^2 c^12+166 a^10 b^4 c^12+86 a^8 b^6 c^12+78 a^6 b^8 c^12+63 a^4 b^10 c^12+52 a^2 b^12 c^12+8 b^14 c^12+20 a^10 b^2 c^14-69 a^8 b^4 c^14-91 a^6 b^6 c^14-83 a^4 b^8 c^14-28 a^2 b^10 c^14+8 b^12 c^14+132 a^10 c^16+109 a^8 b^2 c^16+114 a^6 b^4 c^16+103 a^4 b^6 c^16+50 a^2 b^8 c^16+5 b^10 c^16-165 a^8 c^18-220 a^6 b^2 c^18-206 a^4 b^4 c^18-135 a^2 b^6 c^18-45 b^8 c^18+110 a^6 c^20+167 a^4 b^2 c^20+138 a^2 b^4 c^20+56 b^6 c^20-44 a^4 c^22-61 a^2 b^2 c^22-32 b^4 c^22+10 a^2 c^24+9 b^2 c^24-c^26) : : (barys)

 

= (10 R^4-30 R^2 SB-30 R^2 SC+22 R^2 SW+8 SB SW+8 SC SW-8 SW^2)S^4 + (180 R^8-15 R^6 SB-15 R^6 SC+50 R^4 SB SC-189 R^6 SW+14 R^4 SB SW+14 R^4 SC SW-38 R^2 SB SC SW+60 R^4 SW^2-2 R^2 SB SW^2-2 R^2 SC SW^2+8 SB SC SW^2-6 R^2 SW^3)S^2 + 100 R^8 SB SC-139 R^6 SB SC SW+64 R^4 SB SC SW^2-10 R^2 SB SC SW^3 : : (barys)

 

= lies on this line: {1154,10126}

 

=(6-9-13)-search numbers [-9.9834383081999362654, 13.0351103386165420758, -0.7759019179655640577]

 

Best regards,

Ercole Suppa