[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of H.
Denote:
A"B"C" = the reflection triangle
(ie A", B", C" = the reflections of A,B,C in BC, CA, AB, resp.)
Ab, Ac = the orthogonal projections of A" on BB', CC', resp.
Ma = the midpoint of AbAc. Similarly Mb, Mc.
Denote:
A"B"C" = the reflection triangle
(ie A", B", C" = the reflections of A,B,C in BC, CA, AB, resp.)
Ab, Ac = the orthogonal projections of A" on BB', CC', resp.
Ma = the midpoint of AbAc. Similarly Mb, Mc.
1. The perpendiculars to AbAc, BcBa, CaCb at Ma, Mb, Mc, resp. (perp. bisectors) are concurrent.
2. The centroid of MaMbMc lies on the Euler line of ABC.
2. The centroid of MaMbMc lies on the Euler line of ABC.
3. ABC, MaMbMc are orthologic.
4. A'B'C', MaMbMc are orthologic
5. A"B"C", MaMbMc are orthologic.
[Peter Moses]:
Hi Antreas,
1). X(10263).
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2). X(428).
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3a). ABC, MaMbMc: X(3519).
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3b). MaMbMc ABC: A point noted in Hyacinthos #26047 [1b). NaNbNc ABC] with searches {-5.65106782232459383390229978998,-7.39525863228642826056813410806,11.3686440684094112068512024650}.
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4a). A’B’C’, MaMbMc:
(2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6)::
on lines {{4,539},{6,17},{54,3523},{140,1493},{185,550},{193,10539},{1858,9957},{2888,5068},{3533,6689},{3574,3850},{5073,5895},{5446,11817},{8550,12363}}.
reflection of X(i) in X(j) for these {i,j}: {{1209, 195}, {3519, 12242}, {10625, 11577}, {12325, 6689}}.
2 X[140] - 3 X[1493], 9 X[1209] - 10 X[1656], 9 X[195] - 5 X[1656], 5 X[1656] - 3 X[3519], 3 X[1209] - 2 X[3519], 3 X[195] - X[3519], 9 X[54] - 7 X[3523], 9 X[3574] - 8 X[3850], 9 X[2888] - 13 X[5068], 17 X[3533] - 18 X[6689], 2 X[550] - 3 X[10619], X[4] + 3 X[11271], 3 X[3574] - 4 X[11803], 2 X[3850] - 3 X[11803], 5 X[1656] - 6 X[12242], 3 X[1209] - 4 X[12242], 3 X[195] - 2 X[12242], X[5073] - 9 X[12316], 17 X[3533] - 9 X[12325].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195,3519,12242),(3519,12242, 1209).
on Feurbach of the orthic triangle.
X(4)-Ceva conjugate of X(140).
Searches: {-45. 3536947132350053640870412116, 33. 9753984009394371100020777307, 1. 05170930275015588731093646355} .
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4b). MaMbMc A’B’C’: X(10263).
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5a). A’’B’’C’’, MaMbMc:
5 a^10-20 a^8 b^2+32 a^6 b^4-26 a^4 b^6+11 a^2 b^8-2 b^10-20 a^8 c^2+29 a^6 b^2 c^2-a^4 b^4 c^2-14 a^2 b^6 c^2+6 b^8 c^2+32 a^6 c^4-a^4 b^2 c^4+6 a^2 b^4 c^4-4 b^6 c^4-26 a^4 c^6-14 a^2 b^2 c^6-4 b^4 c^6+11 a^2 c^8+6 b^2 c^8-2 c^10::
on lines {{3,2889},{4,12175},{6,17},{140,12325},{382,539},{1154,1657},{1493,3526},{2888,3851},{7517,11061},{10605,10619}}.
reflection of X(3) in X(11271).
5 X[195] - 4 X[1209], 6 X[195] - 5 X[1656], 6 X[1209] - 5 X[3519], 5 X[1656] - 4 X[3519], 3 X[195] - 2 X[3519], 8 X[1493] - 7 X[3526], 6 X[2888] - 7 X[3851], 7 X[3851] - 8 X[11803], 3 X[2888] - 4 X[11803], 15 X[1656] - 16 X[12242], 9 X[1209] - 10 X[12242], 9 X[195] - 8 X[12242], 3 X[3519] - 4 X[12242], 4 X[10619] - 3 X[12307], 2 X[4] - 3 X[12316], 4 X[140] - 3 X[12325].
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195,3519,1656),(2888,11803, 3851).
on K119.
Searches: {-123. 941135855538668941675950464, 89. 0602209880973423478606901646,- 0. 812810422680549164627067874004 }.
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5b). MaMbMc A’’B’’C’’:
a^2 (3 a^12 b^2-12 a^10 b^4+15 a^8 b^6-15 a^4 b^10+12 a^2 b^12-3 b^14+3 a^12 c^2-22 a^10 b^2 c^2+34 a^8 b^4 c^2-13 a^6 b^6 c^2+10 a^4 b^8 c^2-25 a^2 b^10 c^2+13 b^12 c^2-12 a^10 c^4+34 a^8 b^2 c^4-16 a^6 b^4 c^4+5 a^4 b^6 c^4+10 a^2 b^8 c^4-21 b^10 c^4+15 a^8 c^6-13 a^6 b^2 c^6+5 a^4 b^4 c^6+6 a^2 b^6 c^6+11 b^8 c^6+10 a^4 b^2 c^8+10 a^2 b^4 c^8+11 b^6 c^8-15 a^4 c^10-25 a^2 b^2 c^10-21 b^4 c^10+12 a^2 c^12+13 b^2 c^12-3 c^14)::
on lines {{51,12291},{140,6153},{389,6152},{511,3519},{539,11819},{550,13368},{5446,11803},{6000,6242},{6759,12175},{9935,12234},{9969,11808},{11793,12226},{12380,13367}}.
Reflection of X(i) in X(j) for these {i,j}: {{389, 6152}, {12226, 11793}}.
2 X[140] - 3 X[6153], 3 X[389] - 2 X[10619], 3 X[6152] - X[10619], 3 X[5446] - 2 X[11803], 3 X[11808] - 2 X[12242], 3 X[51] - X[12291], X[550] - 3 X[13368].
Searches: {21. 9444019850531588882082929348,- 29. 7002514145371303890993831056, 14. 0741914681009345010141012826}.
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Best regards,
Peter Moses.
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