[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the antipedal triangle of O.
A"B"C", N1N2N3, M1M2M3 share the same Euler line L.
2.1
A point on Euler X[2] + t X[3] -> a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 2*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 + 3*a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 6*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 - 3*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8)*t : :
G -> X(18281).
O -> X(11250).
H -> H.
N -> X(10224).
Let's see if we can get one point at least!
X(20) ->
= 3 X[2] - 4 X[10226], 5 X[3] - 4 X[10020], 4 X[3] - 3 X[10201], 3 X[4] - 4 X[10224], 3 X[376] - 2 X[1658], 15 X[631] - 16 X[10212], 5 X[631] - 4 X[13406], 7 X[3090] - 8 X[5498], 5 X[3522] - 4 X[15331], 9 X[3524] - 8 X[10125], 3 X[3534] - X[7387], 2 X[5449] - 3 X[11204], 5 X[8567] - 3 X[14852], 3 X[8703] - 2 X[13383], 16 X[10020] - 15 X[10201], 4 X[10212] - 3 X[13406], 2 X[10224] - 3 X[11250], 3 X[10606] - X[12293], 3 X[14070] - 5 X[15696]
= lies on these lines: {2, 3}, {68, 32138}, {74, 25738}, {155, 5925}, {156, 5878}, {1147, 2777}, {1204, 16111}, {3357, 17702}, {4299, 8144}, {4302, 9627}, {4549, 10627}, {4846, 32046}, {5449, 11204}, {5654, 11744}, {5663, 12118}, {6146, 20725}, {7728, 25487}, {8567, 14852}, {10539, 16163}, {10605, 12370}, {10606, 12293}, {10733, 23294}, {11438, 12897}, {12038, 22802}, {12041, 26937}, {12121, 12825}, {12289, 13445}, {13340, 18442}, {13403, 18952}, {13491, 19467}, {14216, 30522}, {14677, 18917}, {15055, 26917}, {15311, 32139}, {17854, 20127}, {18931, 22979}
= midpoint of X(i) and X(j) for these {i,j}: {155, 5925}, {1657, 12085}, {3529, 14790}, {12118, 20427}
= reflection of X(i) in X(j) for these {i,j}: {4, 11250}, {26, 550}, {68, 32138}, {382, 13371}, {5878, 156}, {7728, 25487}, {22802, 12038}, {32140, 3357}
2.2
A point on Euler X[2] + t X[3] -> 4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 8*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 + 3*a^2*(2*a^8 - 4*a^6*b^2 + 4*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 6*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + b^6*c^2 - 3*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*c^6 + b^2*c^6 - 2*c^8)*t : :
G ->
= lies on these lines: {2, 3}, {68, 17821}, {511, 10182}, {524, 1147}, {539, 32391}, {541, 2883}, {542, 10282}, {597, 5462}, {1177, 15132}, {3564, 23041}, {3580, 11464}, {5063, 31406}, {5562, 5642}, {5944, 31804}, {6102, 15361}, {6247, 20191}, {7689, 16252}, {8262, 8550}, {8981, 11266}, {9140, 34224}, {9730, 13394}, {10168, 11695}, {10169, 11649}, {10192, 13754}, {10264, 20773}, {10575, 15738}, {11265, 13966}, {11430, 32223}, {11454, 32111}, {11645, 20299}, {13292, 19357}, {13352, 32269}, {13367, 32225}, {13567, 18475}, {13884, 18459}, {13937, 18457}, {14915, 23328}, {15360, 34148}, {15462, 16789}, {18917, 26864}, {20302, 23358}, {20397, 32274}, {28708, 33878}
= midpoint of X(i) and X(j) for these {i,j}: {2, 14070}, {549, 10154}
O -> X(15331).
H -> X(15761).
N -> X(18282).
Best regards,
Peter Moses.
Tangential triangle version:
Denote:
Na, Nb, Nc = the NPC centers of OB'C', OC'A', OA'B', resp.
N1, N2, N3 = the reflections of Na,Nb,Nc in A'O, B'O, C'O, resp.
MaMbMc = the medial triangle of NaNbNc
M1, M2, M3 = the reflections of Ma, Mb, Mc in A'O, B'O, C'O, resp.
L = the Euler line of ABC
A"B"C" = the reflection of ABC in L
Na, Nb, Nc = the NPC centers of OB'C', OC'A', OA'B', resp.
N1, N2, N3 = the reflections of Na,Nb,Nc in A'O, B'O, C'O, resp.
MaMbMc = the medial triangle of NaNbNc
M1, M2, M3 = the reflections of Ma, Mb, Mc in A'O, B'O, C'O, resp.
L = the Euler line of ABC
A"B"C" = the reflection of ABC in L
A"B"C", N1N2N3, M1M2M3 share the same Euler line L.
1. A"B"C", N1N2N3, M1M2M3 are homothetic.
Which are the homothetic centers of
Which are the homothetic centers of
1.1. A"B"C", N1N2N3,
1.2. N1N2N3, M1M2M3,
1.3. M1M2M3, A"B"C" ?
2.
2.1. Which points wrt triangle ABC are O, H, etc...... of N1N2N3 ?
2.2. Which points wrt triangle ABC are O, H, etc..... of M1M2M3 ?
[|Peter Moses]:
Hi Antreas,
1.1 X(4).
1.2 X(26).
1.2 X(7488).
1.2. N1N2N3, M1M2M3,
1.3. M1M2M3, A"B"C" ?
2.
2.1. Which points wrt triangle ABC are O, H, etc...... of N1N2N3 ?
2.2. Which points wrt triangle ABC are O, H, etc..... of M1M2M3 ?
[|Peter Moses]:
Hi Antreas,
1.1 X(4).
1.2 X(26).
1.2 X(7488).
2.1
A point on Euler X[2] + t X[3] -> a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 2*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 + 3*a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 6*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 - 3*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8)*t : :
G -> X(18281).
O -> X(11250).
H -> H.
N -> X(10224).
Let's see if we can get one point at least!
X(20) ->
= 3*a^10 - 5*a^8*b^2 - 2*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10 - 5*a^8*c^2 + 16*a^6*b^2*c^2 - 8*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 8*a^4*b^2*c^4 + 14*a^2*b^4*c^4 - 2*b^6*c^4 + 6*a^4*c^6 - 6*a^2*b^2*c^6 - 2*b^4*c^6 - a^2*c^8 + 3*b^2*c^8 - c^10 : :
= 3 X[2] - 4 X[10226], 5 X[3] - 4 X[10020], 4 X[3] - 3 X[10201], 3 X[4] - 4 X[10224], 3 X[376] - 2 X[1658], 15 X[631] - 16 X[10212], 5 X[631] - 4 X[13406], 7 X[3090] - 8 X[5498], 5 X[3522] - 4 X[15331], 9 X[3524] - 8 X[10125], 3 X[3534] - X[7387], 2 X[5449] - 3 X[11204], 5 X[8567] - 3 X[14852], 3 X[8703] - 2 X[13383], 16 X[10020] - 15 X[10201], 4 X[10212] - 3 X[13406], 2 X[10224] - 3 X[11250], 3 X[10606] - X[12293], 3 X[14070] - 5 X[15696]
= lies on these lines: {2, 3}, {68, 32138}, {74, 25738}, {155, 5925}, {156, 5878}, {1147, 2777}, {1204, 16111}, {3357, 17702}, {4299, 8144}, {4302, 9627}, {4549, 10627}, {4846, 32046}, {5449, 11204}, {5654, 11744}, {5663, 12118}, {6146, 20725}, {7728, 25487}, {8567, 14852}, {10539, 16163}, {10605, 12370}, {10606, 12293}, {10733, 23294}, {11438, 12897}, {12038, 22802}, {12041, 26937}, {12121, 12825}, {12289, 13445}, {13340, 18442}, {13403, 18952}, {13491, 19467}, {14216, 30522}, {14677, 18917}, {15055, 26917}, {15311, 32139}, {17854, 20127}, {18931, 22979}
= midpoint of X(i) and X(j) for these {i,j}: {155, 5925}, {1657, 12085}, {3529, 14790}, {12118, 20427}
= reflection of X(i) in X(j) for these {i,j}: {4, 11250}, {26, 550}, {68, 32138}, {382, 13371}, {5878, 156}, {7728, 25487}, {22802, 12038}, {32140, 3357}
2.2
A point on Euler X[2] + t X[3] -> 4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 8*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 + 3*a^2*(2*a^8 - 4*a^6*b^2 + 4*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 6*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + b^6*c^2 - 3*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*c^6 + b^2*c^6 - 2*c^8)*t : :
G ->
= 4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 8*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :
= X[3] + 2 X[13383], X[5] + 2 X[1658], X[5] - 4 X[10020], X[26] + 2 X[140], X[68] + 5 X[17821], X[549] - 4 X[15330], X[550] - 4 X[15331], 5 X[631] + X[7387], 5 X[632] - 8 X[10125], 35 X[632] - 32 X[12043], 5 X[632] + 4 X[12107], 5 X[632] - 2 X[13371], X[1658] + 2 X[10020], 7 X[3523] - X[12085], 7 X[3526] - X[14790], 4 X[3530] - X[12084], X[3627] - 4 X[13406], 7 X[3857] - 16 X[12010], 3 X[5054] + X[9909], X[6247] - 4 X[20191], X[7689] + 2 X[16252], 7 X[10125] - 4 X[12043], 2 X[10125] + X[12107], 4 X[10125] - X[13371], X[10154] + 4 X[15330], 3 X[10245] + 5 X[15694], X[10264] + 2 X[20773], 2 X[10282] + X[12359], 8 X[12043] + 7 X[12107], 16 X[12043] - 7 X[13371], 2 X[12107] + X[13371]
= lies on these lines: {2, 3}, {68, 17821}, {511, 10182}, {524, 1147}, {539, 32391}, {541, 2883}, {542, 10282}, {597, 5462}, {1177, 15132}, {3564, 23041}, {3580, 11464}, {5063, 31406}, {5562, 5642}, {5944, 31804}, {6102, 15361}, {6247, 20191}, {7689, 16252}, {8262, 8550}, {8981, 11266}, {9140, 34224}, {9730, 13394}, {10168, 11695}, {10169, 11649}, {10192, 13754}, {10264, 20773}, {10575, 15738}, {11265, 13966}, {11430, 32223}, {11454, 32111}, {11645, 20299}, {13292, 19357}, {13352, 32269}, {13367, 32225}, {13567, 18475}, {13884, 18459}, {13937, 18457}, {14915, 23328}, {15360, 34148}, {15462, 16789}, {18917, 26864}, {20302, 23358}, {20397, 32274}, {28708, 33878}
= midpoint of X(i) and X(j) for these {i,j}: {2, 14070}, {549, 10154}
O -> X(15331).
H -> X(15761).
N -> X(18282).
Best regards,
Peter Moses.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου