HYACINTHOS 24025

 

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

A'b, A'c = the orthogonal projections of A on the external bisectors of B, C (ie on BIa, CIa, resp.)

L'1 = the Euler line of AA'bA'c. Similarly L'2, L'3

L'1, L'2, L'3 are concurrent at a point F'

(they pass through the exFeuerbach points Fa, Fb, Fc, resp.)

The parallels to L'1, L'2, L'3 through:

[...]

5. A1, B1, C1 (where A1, B1, C1 are the orthogonal projections of the excenters Ia, Ib, Ic, resp.  on the sides B'C', C'A', A'B' of the pedal triangle of I)

 

6. A2, B2, C2 (where A2, B2, C2 are the orthogonal projections of the vertices A', B', C', resp. of the pedal triangle of I on the sides IbIc, IcIa, IaIb of the excentral triangle)

are concurrent (I think !).

 

APH

 

[Angel Montesdeoca]:

5. The parallels to L'1, L'2, L'3 through A1, B1, C1 (where A1, B1, C1 are the orthogonal projections of the excenters Ia, Ib, Ic, resp.   on the sides B'C', C'A', A'B' of the pedal triangle of I) are concurrent in
W5 =  (r+2R) X(1) - r X(21) = (r+4R) X(7) - (r+2R) X(79).

W5 = (a (a^4 (b-c)^2-a^5 (b+c)+(b^2-c^2)^2 (b^2-b c+c^2)-a (b-c)^2 (b^3+4 b^2 c+4 b c^2+c^3)+a^3 (2 b^3+3 b^2 c+3 b c^2+2 c^3)+a^2 (-2 b^4+3 b^3 c+6 b^2 c^2+3 b c^3-2 c^4)):...:...)

with (6-9-13)-search number  (1.52086596235072, 1.62091170244161, 1.81655670528601).

W5 is   the midpoint of X(i) and X(j) for these {i,j}:  {3647,3874}.

W5 lies on lines X(i)X(j) for {i,j}: {1,21}, {7,79}, {20,5441}, {27,1844}, {30,553}, {46,7675}, {57,3651}, {65,4304}, {72,5325}, {142,442}, {226,6841}, {354,946}, {377,5883}, {938,2475}, {1012,5884}, {1100,3284}, {1387,2771}, {1697,8000}, {1699,9960}, {1729,2280}, {2646,5427}, {3085,5686}, {3336,7411}, {3555,5837}, {3584,4015}, {3648,9965}, {3833,4197}, {4313,5903}, {5044,6675}, {5249,6701}, {5273,5904}, {5570,6744}, {5719,10021}, {5735,7671}, {5836,8261}.


6. The parallels to L'1, L'2, L'3 through A2, B2, C2 (where A2, B2, C2 are the orthogonal projections of the vertices A', B', C', resp. of the pedal triangle of I on the sides IbIc, IcIa, IaIb of the excentral triangle) are concurrent in
W6 =  (r+3R)X(21) - (r+4R)X(142) = (2r+R) X(35)   - (2r+5R) X(79).

W6 = (2 a^7-a^6 (b+c)-(b-c)^4 (b+c)^3+5 a b c (b^2-c^2)^2+a^3 (b+c)^2 (2 b^2-3 b c+2 c^2)-a^5 (4 b^2+6 b c+4 c^2)+a^4 (b^3-4 b^2 c-4 b c^2+c^3)+a^2 (b-c)^2 (b^3+6 b^2 c+6 b c^2+c^3):...:...)

with (6-9-13)-search number  (-1.13150673357715, -1.02446396734287, 4.87214264402659).

W6 is the midpoint of X(i) and X(j) for these {i,j}:  {79,1770}.

W6 lies on lines X(i)X(j) for {i,j}: {4, 1768}, {9, 3648}, {21, 142}, {30, 553}, {35, 79}, {63, 2475}, {191, 3474}, {442, 1155}, {516, 3649}, {1708, 7701}, {1836, 5248}, {3874, 7354}, {3911, 6841}, {5122, 10021}, {5325, 6175}, {5441, 5557}

 
   Angel Montesdeoca

div>