Κυριακή, 16 Οκτωβρίου 2016

TRIANGLE CENTERS FROM HYACINTHOS H011 - H020

H011 = HATZIPOLAKIS - MOSES

Barycentrics 2 a^9-2 a^8 b-3 a^7 b^2+a^6 b^3+a^5 b^4+5 a^4 b^5-a^3 b^6-5 a^2 b^7+a b^8+b^9-2 a^8 c+8 a^7 b c-a^6 b^2 c-2 a^5 b^3 c-3 a^4 b^4 c-12 a^3 b^5 c+9 a^2 b^6 c+6 a b^7 c-3 b^8 c-3 a^7 c^2-a^6 b c^2+2 a^5 b^2 c^2-2 a^4 b^3 c^2+a^3 b^4 c^2+19 a^2 b^5 c^2-16 a b^6 c^2+a^6 c^3-2 a^5 b c^3-2 a^4 b^2 c^3+24 a^3 b^3 c^3-23 a^2 b^4 c^3-6 a b^5 c^3+8 b^6 c^3+a^5 c^4-3 a^4 b c^4+a^3 b^2 c^4-23 a^2 b^3 c^4+30 a b^4 c^4-6 b^5 c^4+5 a^4 c^5-12 a^3 b c^5+19 a^2 b^2 c^5-6 a b^3 c^5-6 b^4 c^5-a^3 c^6+9 a^2 b c^6-16 a b^2 c^6+8 b^3 c^6-5 a^2 c^7+6 a b c^7+a c^8-3 b c^8+c^9::

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

Denote:

Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.

M1, M2, M3 = the midpoints of IA', IB', IC', resp.

MaMbMc, M1M2M3 are cyclologic. The point is the cyclologic center (MaMbMc, M1M2M3). The other cyclologic center (M1M2M3, MaMbMc) is the point X(1387)

(Antreas Hatzipolakis and Peter Moses, Sept. 20, 2016. See: Hyacinthos #24436)

The point lies on these lines: {1,1537},{55,108},{123,3816}, ...

H012 = HATZIPOLAKIS - MOSES

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

3 X[3] + X[146], 3 X[5] - X[265], 3 X[110] + X[265], 3 X[2] + X[399], X[74] - 3 X[549], 3 X[113] - X[1539], 3 X[1511] + X[1539], 5 X[1656] - X[3448], 2 X[3628] + X[5609], X[1511] - 3 X[5642], X[113] + 3 X[5642], X[1539] + 9 X[5642], X[2931] + 3 X[5654], X[74] + 3 X[5655], X[2948] + 3 X[5886], 3 X[5972] + X[6053], 3 X[140] + 2 X[6053], 3 X[140] - 2 X[6699], 3 X[5972] - X[6699], 3 X[5066] - 2 X[7687], 3 X[5055] + X[9143], 3 X[597] - X[9976].

Let ABC be a triangle, NaNbNc the pedal triangle of N and OaObOc the pedal triangle of O.

Denote:

N1, N2, N3 = the reflections of N in BC, CA, AB, resp.

O1,O2, O3 = the reflections of O in BC, CA, AB, resp.

The point is the intersection of the parallels to O1N1, O2N2, O3N3 through Na,Nb,Nc, resp. The parallels to O1N1, O2N2, O3N3 through Oa,Ob,Oc, resp. concur at X(1511). The lines O1N1, O2N2, O3N3 concur at X(110)

(Antreas Hatzipolakis and Peter Moses, Sept. 21, 2016. See: Hyacinthos #24449)

The point lies on these lines:{2,399},{3,146},{4,7666},{5,4 9},{30,113},{69,10201},{74,549 },{125,3628},{140,5663},{403,3 043},{468,1986},{495,10091},{4 96,10088},{542,547},{546,9820} ,{548,2777},{550,7728},{597, 9976},{1125,2771},{1154,10096} ,{1656,3448},{2931,5654},{ 2948,5886},{3564,6593},{3582, 6126},{3584,7343},{3850,10113} ,{5055,9143},{5066,7687},{ 5432,7727},{5876,10125},{5898, 7693},{6140,6592},{6153,10095} ,{6677,9826},{7525,10117},{754 2,7723},{7722,10018} = Complement of the complement of X(399).

= Midpoint of X(i) and X(j) for these {i,j}: {5,110},{113,1511},{125,5609} ,{549,5655},{550,7728},{6053,6 699}}. Reflection of X(i) in X(j) for these {i,j}: {{125,3628},{140,5972},{10113, 3850}.

= Crossdifference of every pair of points on line {2081,2433}.

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113,5642,1511), (5972,6053,6699).

H013 = HATZIPOLAKIS - MOSES

Barycentrics a (3 a^5 b-3 a^4 b^2-6 a^3 b^3+6 a^2 b^4+3 a b^5-3 b^6+3 a^5 c-6 a^4 b c+8 a^3 b^2 c-11 a b^4 c+6 b^5 c-3 a^4 c^2+8 a^3 b c^2-16 a^2 b^2 c^2+8 a b^3 c^2+3 b^4 c^2-6 a^3 c^3+8 a b^2 c^3-12 b^3 c^3+6 a^2 c^4-11 a b c^4+3 b^2 c^4+3 a c^5+6 b c^5-3 c^6)::

(3 r + 2 R) X[1] - (3 r - R) X[3]. 2 X[942]-X[10247],X[3057]-4 X[5885],X[3576]+X[5903],3 X[10202]-2 X[10246],2 X[5]-5 X[4004],X[355]-4 X[10107],X[3817]-3 X[3919],5 X[3698]-2 X[5694],4 X[3754]-X[5887],2 X[3754]-X[10175],X[5887]-2 X[10175],7 X[3922]-4 X[9956],X[4018]+2 X[5690].

Let ABC be a triangle.

Denote:

A', B', C' = the reflections of I in BC, CA, AB, resp.

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the reflections of Na, Nb, Nc in B'C', C'A', A'B', resp.

The point is the centroid of N1N2N3 lying on the OI line of ABC.

(Antreas Hatzipolakis and Peter Moses, Oct. 13, 2016. See: Hyacinthos #24611)

The point lies on these lines: {1,3},{5,4004},{355,10107},{1 864,6797},{2800,3817},{3698,56 94},{3754,5887},{3922,9956},{ 4018,5690},{4323,6961},{4848,6 842},{5927,9952}

= Midpoint of X(3576) and X(5903).

= Reflection of X(i) in X(j) for these {i,j}: {{5887, 10175}, {10175, 3754}, {10247, 942}}.

H014 = HATZIPOLAKIS - LOZADA - MOSES

Barycentrics a^4 (a^12-4 a^10 b^2+5 a^8 b^4-5 a^4 b^8+4 a^2 b^10-b^12-4 a^10 c^2+9 a^8 b^2 c^2-5 a^6 b^4 c^2+a^4 b^6 c^2-3 a^2 b^8 c^2+2 b^10 c^2+5 a^8 c^4-5 a^6 b^2 c^4+2 a^4 b^4 c^4-a^2 b^6 c^4-b^8 c^4+a^4 b^2 c^6-a^2 b^4 c^6-5 a^4 c^8-3 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+2 b^2 c^10-c^12)::

Trilinears cos(2*A)*cos(3*A)-cos(4*A)* cos(B-C) ::

R^2*X(4)+(7*R^2-2*SW)*X(54)

2 X[54] + X[6759], 3 X[154] - X[9920].

X[4] + (J^2 - 2) X[54], (J = OH/R).

Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P. Denote: O', Oa, Ob, Oc = the circumcenters of A'B'C', PBC, PCA, PAB, resp. The reflections of O'Oa, O'Ob, O'Oc in BC, CA, AB, resp. are concurrent for P = O. The point is the point of concurrence for P = O

(Antreas P. Hatzipolakis, Peter Moses, Cesar Lozada, Oct. 8, 2016. See: Hyacinthos #24566 and #24574)

The point lies on these lines:{3,8157},{4,54},{49,52},{110, 2888},{154,9704},{156,9927},{ 182,6689},{206,576},{539, 10201},{569,6145},{1092,7691}, {1147,1154},{1209,6639},{1971, 9697},{2904,9707},{3518,7730}, {6288,10254},{9813,9827},{ 10182,10203}

= Midpoint of X(195) and X(2917)

= X(324)-Ceva conjugate of X(571).

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (54,3574,578).

H015 = HATZIPOLAKIS - MOSES

Barycentrics 14 a^16-103 a^14 b^2+335 a^12 b^4-633 a^10 b^6+765 a^8 b^8-609 a^6 b^10+313 a^4 b^12-95 a^2 b^14+13 b^16-103 a^14 c^2+454 a^12 b^2 c^2-729 a^10 b^4 c^2+342 a^8 b^6 c^2+459 a^6 b^8 c^2-786 a^4 b^10 c^2+469 a^2 b^12 c^2-106 b^14 c^2+335 a^12 c^4-729 a^10 b^2 c^4+360 a^8 b^4 c^4+15 a^6 b^6 c^4+480 a^4 b^8 c^4-837 a^2 b^10 c^4+376 b^12 c^4-633 a^10 c^6+342 a^8 b^2 c^6+15 a^6 b^4 c^6-14 a^4 b^6 c^6+463 a^2 b^8 c^6-758 b^10 c^6+765 a^8 c^8+459 a^6 b^2 c^8+480 a^4 b^4 c^8+463 a^2 b^6 c^8+950 b^8 c^8-609 a^6 c^10-786 a^4 b^2 c^10-837 a^2 b^4 c^10-758 b^6 c^10+313 a^4 c^12+469 a^2 b^2 c^12+376 b^4 c^12-95 a^2 c^14-106 b^2 c^14+13 c^16::

Let A be a triangle.

Denote:

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

Denote:

Nab, Nac = the orthogonal projections of Na on BNb, CNc, resp.

Nbc, Nba = the orthogonal projections of Nb on CNc, ANa, resp.

Nca, Ncb = the orthogonal projections of Nc on ANa, BNb, resp.

Let Oa, Ob, Oc be the circumcenters of NaNabNac, NbNbcNba, NcNcaNc, resp.

ABC, OaObOc are orthologic.

ABC, OaObOc are orthologic at X(1263)

(Antreas Hatzipolakis and Peter Moses, Sept. 30, 2016. See: Hyacinthos #24515)

The point is the orthologic center (OaObOc,ABC)

The point lies on these lines: {140, 930}.

= Midpoint of X(140), X(1487).

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