HYACINTHOS 26744

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the pedal triangle of H.

Denote:

MaMbMc= the midheight triangle
(ie Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.)

Ab, Ac = the orthogonal projections of Ma on AB, AC, resp.
Bc, Ba = the orthogonal projections of Mb on BC, BA, resp.
Ca, Cb = the orthogonal projections of Mc on CA, CB, resp.

La, Lb, Lc = the Euler lines of AAbAc, BBcBa, CCaCb, resp.

1. La, Lb, Lc are concurrent.
2. the parallels to La, Lb, Lc through A, B, C are concurrent.
3. the parallels to La, Lb, Lc through A', B', C' are concurrent.
4. the parallels to La, Lb, Lc through Ma, Mb, Mc are concurrent.

Denote:

L1, L2, L3 = the reflections of La, Lb, Lc in BC, CA, AB, resp.
A*B*C* = the triangle bounded by L1, L2, L3

Li, Lii, Liii = the reflections of La, Lb, Lc in AA', BB', CC', resp.
A**B**C** = the triangle bounded by Li, Lii, Liii

5. A'B'C', A*B*C* are parallelogic.

6. A'B'C', A**B**C** are parallelogic

Parallelogic center (A'B'C', A*B*C*) = Parallelogic center (A'B'C', A**B**C**)

7. A*B*C*, A**B**C** are homothetic (and congruent).
Homothetic center on the Euler line.



[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+4 a^10 b^2 c^2-4 a^8 b^4 c^2-24 a^6 b^6 c^2+41 a^4 b^8 c^2-20 a^2 b^10 c^2+2 b^12 c^2-4 a^10 c^4-4 a^8 b^2 c^4+48 a^6 b^4 c^4-36 a^4 b^6 c^4-4 a^2 b^8 c^4+5 a^8 c^6-24 a^6 b^2 c^6-36 a^4 b^4 c^6+40 a^2 b^6 c^6-b^8 c^6+41 a^4 b^2 c^8-4 a^2 b^4 c^8-b^6 c^8-5 a^4 c^10-20 a^2 b^2 c^10+4 a^2 c^12+2 b^2 c^12-c^14):: 

on lines {{6,74},{67,12317},{125,235},{ 140,5663},{541,9826},{1112, 10990},{1539,7706},{2777, 10110},{3532,11412},{5656, 7505},{6101,7689},{12099, 13202},{12236,14677}}.
midpoint of X(i) and X(j) for these {i,j}: {{74, 974}, {1112, 10990}, {12236, 14677}}.
3 X[974] - X[1986], 3 X[74] + X[1986], 5 X[1986] - 9 X[5890], 5 X[974] - 3 X[5890], 5 X[74] + 3 X[5890], 7 X[1986] - 3 X[7731], 7 X[974] - X[7731], 7 X[74] + X[7731], 5 X[6699] - 3 X[10170], 5 X[125] - X[11381], 4 X[10110] - 5 X[11746], X[6101] - 5 X[12041], 3 X[11381] - 5 X[12133], 3 X[125] - X[12133], 3 X[12099] - X[13202].
{X(74),X(5622)}-harmonic conjugate of X(2935).
crosssum of X(3) and X(6053).
Search: {10. 1299282250599484991025183340, 10. 2637171125009710644941321602,- 8. 14033731523626940265566716252} .

2) X(74).

3) X(1986).

4) X(974).

5)

(A'B'C', A*B*C*): X(403).

(A*B*C*, A'B'C'): 

6 a^10-19 a^8 b^2+20 a^6 b^4-6 a^4 b^6-2 a^2 b^8+b^10-19 a^8 c^2-8 a^6 b^2 c^2+6 a^4 b^4 c^2+24 a^2 b^6 c^2-3 b^8 c^2+20 a^6 c^4+6 a^4 b^2 c^4-44 a^2 b^4 c^4+2 b^6 c^4-6 a^4 c^6+24 a^2 b^2 c^6+2 b^4 c^6-2 a^2 c^8-3 b^2 c^8+c^10:: 

on lines {{403,1503},{468,13399},{1498, 6353},{1596,6759},{6756,14862} ,{6995,12233}}.
3 X[468] - X[13399], X[403] + 3 X[14157], 13 X[403] - 9 X[14644], 13 X[14157] + 3 X[14644].
Search: {-4. 50890493222066827722494123910, -6. 06882130218691805143195888678, 9. 92318919829255388205968490370} .

6)

(A'B'C', A**B**C**):  X(403).

(A**B**C**, A'B'C'):

6 a^10-11 a^8 b^2+4 a^6 b^4-6 a^4 b^6+14 a^2 b^8-7 b^10-11 a^8 c^2+8 a^6 b^2 c^2+6 a^4 b^4 c^2-24 a^2 b^6 c^2+21 b^8 c^2+4 a^6 c^4+6 a^4 b^2 c^4+20 a^2 b^4 c^4-14 b^6 c^4-6 a^4 c^6-24 a^2 b^2 c^6-14 b^4 c^6+14 a^2 c^8+21 b^2 c^8-7 c^10:: 

on lines {{403,1503},{427,11424},{5893, 11457},{6000,11746},{6146, 7577},{11572,11745},{12828, 13202}}.
midpoint of X(i) and X(j) for these {i,j}: {{13399, 13473}}.
3 X[13202] + 5 X[13399], 3 X[13202] - 5 X[13473], X[13202] - 5 X[13851], X[13473] - 3 X[13851], X[13399] + 3 X[13851], 7 X[403] - 3 X[14157], 5 X[403] - 9 X[14644].
{X(13399),X(13851)}-harmonic conjugate of X(13473).

Search: {2. 33600527813314673617733411520, 2. 16261396645662738944401529499, 1. 06531391522218472449834492203} .

7) X(403).


Best regards,
Peter Moses