HYACINTHOS 28835

[Kadir Altintas]:

 

Let ABC be a triangle with orthocenter H

 

Denote:

DEF = the pedal triangle of H (orthic triangle).

D'E'F' = the circumcevian triangle of H

 

The circles with centers D,E,F passing through H intersect the circumcircle of ABC at A',B',C', resp. (other than D',E',F', resp)

Circles are tangent to these circles externally in pairs and to circumcircle of ABC internally at A'',B'',C'', resp.

 

 

Prove:

1. D'A'', E'B'', F'C'' concur on the Euler line of ABC

2. A'A'', B'B'', C'C'' concur on the Euler line of ABC

 

PERU GEOMETICO

 

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[Ercole Suppa]

 

1.  The lines D'A'', E'B'', F'C'' concur on the Euler line of ABC at point:

 

X = REFLECTION OF X(4) in X(431)

 

= a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^8-a^7 b-3 a^6 b^2+3 a^5 b^3+3 a^4 b^4-3 a^3 b^5-a^2 b^6+a b^7-a^7 c-a^6 b c+a^5 b^2 c+a^4 b^3 c+a^3 b^4 c+a^2 b^5 c-a b^6 c-b^7 c-3 a^6 c^2+a^5 b c^2+4 a^4 b^2 c^2-a^2 b^4 c^2-a b^5 c^2+3 a^5 c^3+a^4 b c^3-2 a^2 b^3 c^3+a b^4 c^3+b^5 c^3+3 a^4 c^4+a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-3 a^3 c^5+a^2 b c^5-a b^2 c^5+b^3 c^5-a^2 c^6-a b c^6+a c^7-b c^7) : : (barys) 

 

= (8 a R^4-8 b R^4-4 a R^2 SB+4 b R^2 SB-4 a R^2 SC+4 c R^2 SC-2 a R^2 SW+6 b R^2 SW+a SB SW-b SB SW+a SC SW-c SC SW-b SW^2) S^2 +(-8 R^3 SB SC+2 R SB SC SW) S +2 b R^2 SB SC^2-2 c R^2 SB SC^2-2 b R^2 SB SC SW-b SB SC^2 SW+c SB SC^2 SW+b SB SC SW^2 : : (barys)

 

As a point on the Euler line, X has Shinagawa coefficients: {(p - r - 2 R) (r + R) (p + r + 2 R), r^3 + 6 r^2 R + 10 r R^2 + 6 R^3 - p^2 (r + 2 R)}

= lies on these lines: {2,3}, {100,1299}, {108,1300}, {3563,26706}

 

= reflection of X(4) in X(431) 

 

= (6-8-13) search numbers [3.85528353075841893, 2.97559327256201449, -0.198723259447055052]

 

 

2. The lines A'A'', B'B'', C'C'' concur on the Euler line of ABC at point:

 

Y = EULER LINE INTERCEPT OF X(108)X(1299)

 

= a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^11-3 a^9 b^2+2 a^7 b^4+2 a^5 b^6-3 a^3 b^8+a b^10+a^9 b c-a^8 b^2 c-4 a^7 b^3 c+4 a^6 b^4 c+6 a^5 b^5 c-6 a^4 b^6 c-4 a^3 b^7 c+4 a^2 b^8 c+a b^9 c-b^10 c-3 a^9 c^2-a^8 b c^2+6 a^7 b^2 c^2+2 a^6 b^3 c^2-4 a^5 b^4 c^2+2 a^3 b^6 c^2-2 a^2 b^7 c^2-a b^8 c^2+b^9 c^2-4 a^7 b c^3+2 a^6 b^2 c^3+4 a^5 b^3 c^3+2 a^4 b^4 c^3-6 a^2 b^6 c^3+2 b^8 c^3+2 a^7 c^4+4 a^6 b c^4-4 a^5 b^2 c^4+2 a^4 b^3 c^4+2 a^3 b^4 c^4+4 a^2 b^5 c^4-2 b^7 c^4+6 a^5 b c^5+4 a^2 b^4 c^5-2 a b^5 c^5+2 a^5 c^6-6 a^4 b c^6+2 a^3 b^2 c^6-6 a^2 b^3 c^6-4 a^3 b c^7-2 a^2 b^2 c^7-2 b^4 c^7-3 a^3 c^8+4 a^2 b c^8-a b^2 c^8+2 b^3 c^8+a b c^9+b^2 c^9+a c^10-b c^10)  : : (barys) 

 

As a point on the Euler line, Y has Shinagawa coefficients: {(p - r - 2 R) (p + r + 2 R) (p^2 - r^2 - 2 r R - 2 R^2), -p^4 - r^4 - 8 r^3 R - 22 r^2 R^2 - 28 r R^3 - 12 R^4 + 2 p^2 (r + R) (r + 3 R)} 

= lies on these lines: {2,3}, {108,1299}

 

= (6-8-13) search numbers [-1.34649003052258061, -2.21245787872125866, 5.79382302741721076]

 

 

Best regards

Ercole Suppa