Σάββατο 2 Νοεμβρίου 2019

HYACINTHOS 29563

[Antreas P. Hatzipolakis]:

 
Tangential triangle version:
 
Let ABC be a triangle and A'B'C' the antipedal triangle of O.
 
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

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 
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).

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.

 

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