HYACINTHOS 27110

[Antreas P. Hatzipolakis]:

Let ABC be a triangle, A'B'C' the pedal triangle of Ι and A"B"C" the extouch triangle

ie A', B', C' = the orthogonal projections of the excenters Ia, Ib, Ic on BC, CA, AB, resp.

Denote:

Aa, Ab, Ac = the orthogonal projections of A" on AI:, BI, CI, resp

Ba, Bb, Bc = the orthogonal projections of B" on AI:, BI, CI, resp
Ca, Cb, Cc = the orthogonal projections of C" on AI:, BI, CI, resp

Na, Nb, Nc = the NPC centers of AaAbAc, BaBbBc, CaCbCc, resp.

1. NaNbNc, ABC are orthologic.

2. NaNbNc, A'B'C' are ortologic

3. NaNbNc, A"B"C" are orthologic.

 

4. NaNbNc, IaIbIc are orthologic.

 

[Peter Moses]:


Hi Antreas,

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1a, (NaNbNc,  ABC); 

2 a^7-a^6 b-4 a^5 b^2+a^4 b^3+2 a^3 b^4+a^2 b^5-b^7-a^6 c+8 a^5 b c+7 a^4 b^2 c-8 a^3 b^3 c-7 a^2 b^4 c+b^6 c-4 a^5 c^2+7 a^4 b c^2-12 a^3 b^2 c^2+6 a^2 b^3 c^2+3 b^5 c^2+a^4 c^3-8 a^3 b c^3+6 a^2 b^2 c^3-3 b^4 c^3+2 a^3 c^4-7 a^2 b c^4-3 b^3 c^4+a^2 c^5+3 b^2 c^5+b c^6-c^7:: 

on lines {{3,10624},{5,516},{30,5795},{40,1478},{165,6891},{443,3587},{517,4298},{1788,3359},{2771,6743},{3654,12527},{4292,12702},{5082,7171},{5766,6908},{5853,13369},{6738,13145},{6847,9778},{10310,11375},{11278,12577},{12575,13624}}.
midpoint of X(4292) and X(12702).
reflection of X(i) in X(j) for these {i,j}: {{6738, 13145}, {11278, 12577}, {12575, 13624}}.
3 X[40] + X[1770], 3 X[3] - X[10624], 3 X[3654] - X[12527].

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1b, (ABC, NaNBNc);
(a^5+2 a^4 b-3 a^3 b^2-3 a^2 b^3+2 a b^4+b^5-a^4 c+5 a^3 b c+4 a^2 b^2 c+5 a b^3 c-b^4 c-2 a^3 c^2-3 a^2 b c^2-3 a b^2 c^2-2 b^3 c^2+2 a^2 c^3-5 a b c^3+2 b^2 c^3+a c^4+b c^4-c^5) (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5+2 a^4 c+5 a^3 b c-3 a^2 b^2 c-5 a b^3 c+b^4 c-3 a^3 c^2+4 a^2 b c^2-3 a b^2 c^2+2 b^3 c^2-3 a^2 c^3+5 a b c^3-2 b^2 c^3+2 a c^4-b c^4+c^5):: 

on lines {{9,3652},{30,3680},{2771,12641},{2827,14224},{2829,13143},{3062,7681},{5559,6001},{7091,11373},{12114,15180}}.
on Feuerbach hyperbola.
X(3)-vertex conjugate of X(5559).

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2a, (NaNbNc, A'B'C'); X(4662).

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2b, (A'B'C', NaNbNc); 

(2 a^3+a^2 b-2 a b^2-b^3+a^2 c-2 a b c+b^2 c-2 a c^2+b c^2-c^3) (2 a^4+a^3 b-5 a^2 b^2-a b^3+3 b^4+a^3 c+10 a^2 b c+a b^2 c-5 a^2 c^2+a b c^2-6 b^2 c^2-a c^3+3 c^4):: 

on lines {{7,496},{30,145},{79,5586},{442,10940},{1317,10624},{3305,3652},{3579,3650},{3647,6700},{5815,11684},{6001,13375},{6831,10044}}.
reflection of X(10308) in X(11544).
4 X[3579] - 3 X[3650].

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3a, (NaNbNc A''B"C"); 

a (3 a^8 b-6 a^7 b^2-6 a^6 b^3+18 a^5 b^4-18 a^3 b^6+6 a^2 b^7+6 a b^8-3 b^9+3 a^8 c+6 a^7 b c-2 a^6 b^2 c-18 a^5 b^3 c-12 a^4 b^4 c+18 a^3 b^5 c+18 a^2 b^6 c-6 a b^7 c-7 b^8 c-6 a^7 c^2-2 a^6 b c^2+56 a^5 b^2 c^2-12 a^4 b^3 c^2-14 a^3 b^4 c^2-2 a^2 b^5 c^2-36 a b^6 c^2+16 b^7 c^2-6 a^6 c^3-18 a^5 b c^3-12 a^4 b^2 c^3+28 a^3 b^3 c^3-22 a^2 b^4 c^3+6 a b^5 c^3+24 b^6 c^3+18 a^5 c^4-12 a^4 b c^4-14 a^3 b^2 c^4-22 a^2 b^3 c^4+60 a b^4 c^4-30 b^5 c^4+18 a^3 b c^5-2 a^2 b^2 c^5+6 a b^3 c^5-30 b^4 c^5-18 a^3 c^6+18 a^2 b c^6-36 a b^2 c^6+24 b^3 c^6+6 a^2 c^7-6 a b c^7+16 b^2 c^7+6 a c^8-7 b c^8-3 c^9):: 

on line {{971,3826}}.

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3b, (A''B"C", NaNbNc); 

(2 a^4+a^3 b-5 a^2 b^2-a b^3+3 b^4+a^3 c+10 a^2 b c+a b^2 c-5 a^2 c^2+a b c^2-6 b^2 c^2-a c^3+3 c^4) (4 a^6-3 a^5 b-9 a^4 b^2+6 a^3 b^3+6 a^2 b^4-3 a b^5-b^6-3 a^5 c+10 a^4 b c+10 a^3 b^2 c-12 a^2 b^3 c-7 a b^4 c+2 b^5 c-9 a^4 c^2+10 a^3 b c^2-12 a^2 b^2 c^2+10 a b^3 c^2+b^4 c^2+6 a^3 c^3-12 a^2 b c^3+10 a b^2 c^3-4 b^3 c^3+6 a^2 c^4-7 a b c^4+b^2 c^4-3 a c^5+2 b c^5-c^6):: 

on line {{9,3652}}.
on Mandart hyperbola.
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4a, (NaNbNc, IaIbIc); X(4662).

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4b, (IaIbIc, NaNbNc); 

a (a^9-a^8 b-4 a^7 b^2+4 a^6 b^3+6 a^5 b^4-6 a^4 b^5-4 a^3 b^6+4 a^2 b^7+a b^8-b^9-a^8 c+9 a^7 b c+8 a^6 b^2 c-29 a^5 b^3 c-16 a^4 b^4 c+31 a^3 b^5 c+12 a^2 b^6 c-11 a b^7 c-3 b^8 c-4 a^7 c^2+8 a^6 b c^2+42 a^5 b^2 c^2+10 a^4 b^3 c^2-20 a^3 b^4 c^2-24 a^2 b^5 c^2-18 a b^6 c^2+6 b^7 c^2+4 a^6 c^3-29 a^5 b c^3+10 a^4 b^2 c^3-14 a^3 b^3 c^3+8 a^2 b^4 c^3+11 a b^5 c^3+10 b^6 c^3+6 a^5 c^4-16 a^4 b c^4-20 a^3 b^2 c^4+8 a^2 b^3 c^4+34 a b^4 c^4-12 b^5 c^4-6 a^4 c^5+31 a^3 b c^5-24 a^2 b^2 c^5+11 a b^3 c^5-12 b^4 c^5-4 a^3 c^6+12 a^2 b c^6-18 a b^2 c^6+10 b^3 c^6+4 a^2 c^7-11 a b c^7+6 b^2 c^7+a c^8-3 b c^8-c^9) 

on lines {{1,10308},{9,3652},{30,2136},{40,3650},{631,5506},{2800,13144},{2950,6223},{3646,7701},{5541,6361}}.
on Jerabek hyperbola of the excentral triangle.
X(3650)-zayin conjugate of X(40).

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Best regards,
Peter Moses.