[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the pedal triangle of N.
Denote:
A"B"C" = the pedal triangle of N wrt triangle A'B'C'.
La, Lb, Lc = The Euler lines of A'B"C", B'C"A", C'A"B", resp.
Denote:
A"B"C" = the pedal triangle of N wrt triangle A'B'C'.
La, Lb, Lc = The Euler lines of A'B"C", B'C"A", C'A"B", resp.
1. La, Lb, Lc are concurrent.
2. The parallels to La, Lb, Lc through A', B', C', resp. are concurrent.
2. The parallels to La, Lb, Lc through A', B', C', resp. are concurrent.
[Peter Moses]:
Hi Antreas,
1). 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^2-6 a^10 b^2 c^2+6 a^8 b^4 c^2+a^6 b^6 c^2+6 a^4 b^8 c^2-15 a^2 b^10 c^2+7 b^12 c^2-4 a^10 c^4+6 a^8 b^2 c^4-4 a^6 b^4 c^4-5 a^4 b^6 c^4+22 a^2 b^8 c^4-15 b^10 c^4+5 a^8 c^6+a^6 b^2 c^6-5 a^4 b^4 c^6-22 a^2 b^6 c^6+9 b^8 c^6+6 a^4 b^2 c^8+22 a^2 b^4 c^8+9 b^6 c^8-5 a^4 c^10-15 a^2 b^2 c^10-15 b^4 c^10+4 a^2 c^12+7 b^2 c^12-c^14)::
on lines {{5,6153},{30,5462},{51,3153}, {182,5899},{186,5943},{511, 2072},...}.
midpoint of X(i) and X(j) for these {i,j}: {{5,11692},{1568,11800}}.
3 X[51] + X[3153], X[186] - 3 X[5943].
Searches: {0. 310635902977653405216625022515 ,-0. 114914211941931691727354512037 ,3. 57685005803141485652192821583} .
2). X(11692).
Best regards,
Peter Moses.
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