[APH]: Let ABC be a triangle and A'B'C' the pedal triangle of H.Denote:Bc = the reflection of B' in HC', Cb = the reflection of C' in HB'
Ca = the reflection of C' in HA',
Ac = the reflection of A' in HC'
Ab = the reflection of A' in HB',
Ba = the reflection of B' in HA'
(BcCb, CaAc, AbBa are perpendiculars to Euler line of ABC)
H1, H2, H3 = the orthogonal projections of A, B, C on BcCb, CaAc, AbBa, resp.
1. ABC, H1H2H3 are orthologic
2. The centroid of H1H2H3 lies on the Euler line of ABC.
[Peter Moses]:
Hi Antreas,
1)
(ABC, H1H2H3): (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14-a^12 c^2+5 a^10 b^2 c^2-8 a^8 b^4 c^2+13 a^4 b^8 c^2-13 a^2 b^10 c^2+4 b^12 c^2+3 a^10 c^4-3 a^8 b^2 c^4+12 a^6 b^4 c^4-20 a^4 b^6 c^4+13 a^2 b^8 c^4-5 b^10 c^4-2 a^8 c^6-6 a^6 b^2 c^6+12 a^4 b^4 c^6-2 a^6 c^8-3 a^4 b^2 c^8-8 a^2 b^4 c^8+5 b^6 c^8+3 a^4 c^10+5 a^2 b^2 c^10-4 b^4 c^10-a^2 c^12+b^2 c^12) (a^12 b^2-3 a^10 b^4+2 a^8 b^6+2 a^6 b^8-3 a^4 b^10+a^2 b^12-a^12 c^2-5 a^10 b^2 c^2+3 a^8 b^4 c^2+6 a^6 b^6 c^2+3 a^4 b^8 c^2-5 a^2 b^10 c^2-b^12 c^2+4 a^10 c^4+8 a^8 b^2 c^4-12 a^6 b^4 c^4-12 a^4 b^6 c^4+8 a^2 b^8 c^4+4 b^10 c^4-5 a^8 c^6+20 a^4 b^4 c^6-5 b^8 c^6-13 a^4 b^2 c^8-13 a^2 b^4 c^8+5 a^4 c^10+13 a^2 b^2 c^10+5 b^4 c^10-4 a^2 c^12-4 b^2 c^12+c^14):: on lines {{26,15454},{14254,15761}}.
(H1H2H3, ABC): a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-8 a^10 b^2 c^2+8 a^8 b^4 c^2+8 a^6 b^6 c^2-7 a^4 b^8 c^2-8 a^2 b^10 c^2+6 b^12 c^2-4 a^10 c^4+8 a^8 b^2 c^4-28 a^6 b^4 c^4+12 a^4 b^6 c^4+24 a^2 b^8 c^4-12 b^10 c^4+5 a^8 c^6+8 a^6 b^2 c^6+12 a^4 b^4 c^6-40 a^2 b^6 c^6+7 b^8 c^6-7 a^4 b^2 c^8+24 a^2 b^4 c^8+7 b^6 c^8-5 a^4 c^10-8 a^2 b^2 c^10-12 b^4 c^10+4 a^2 c^12+6 b^2 c^12-c^14):: 4 X[389] - 3 X[974], 2 X[389] - 3 X[1112], 3 X[1539] - X[5876], 3 X[1986] - X[6241], X[6243] + 3 X[7728], 3 X[74] - 7 X[9781], 3 X[113] - X[10625], X[6241] + 3 X[10721], 3 X[51] - X[10990], 7 X[9781] - 6 X[11746], X[974] - 4 X[11807], X[389] - 3 X[11807], 4 X[10110] - 3 X[12099], X[20] - 3 X[12824], X[11381] - 3 X[13202], 3 X[5972] - 2 X[13348], X[11381] + 3 X[13417], 3 X[13417] - X[14448], 3 X[13202] + X[14448], 3 X[11557] - X[14641], 9 X[5640] - 5 X[15021], 3 X[12041] - 5 X[15026], 9 X[7998] - 13 X[15029], 13 X[15028] - 9 X[15055], 3 X[125] - 4 X[15465].
on lines {{4,67},{20,12824},{51,10990},{52,3627},{74,9781},{113,10625},{125,1595},{389,974},{511,1514},{541,5446},{542,13598},{973,13488},{1192,2935},{1539,5876},{1986,5895},{2854,10752},{3542,15131},{5198,15106},{5622,10982},{5640,15021},{5972,13348},{6000,13148},{6243,7728},{6593,12082},{7530,15132},{7731,12292},{7998,15029},{9707,15463},{9919,11426},{10110,12099},{10117,11425},{10628,11576},{11414,15462},{11557,14641},{12041,15026},{12244,15151},{14984,15063},{15028,15055}}. midpoint of X(i) and X(j) for these {i,j}: {{1986, 10721}, {7731, 12292}, {11381, 14448}, {13202, 13417}}. reflection of X(i) in X(j) for these {i,j}: {{74, 11746}, {974, 1112}, {1112, 11807}, {12244, 15151}, {15738, 4}}. {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11381, 13417, 14448), (13202, 14448, 11381). crosssum of X(3) and X(16003). 2) X(428). Best regards, Peter.
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