HYACINTHOS 26259

The foci of the Steiner inellipse of a triangle are the intersections of the inellipse's major axis and the circle with center on the minor axis and going through the Fermat points.

 

Benedetto Scimemi, Simple Relations Regarding the Steiner Inellipse of a Triangle 
FG, p.74, theorem 6

Which is the center of the circle?

 

APH

[César Lozada]:

 

Center = Tripolar centroid of X(3413)

= (SB-SC)^2*(SA^2-SB*SC+sqrt(-3* S^2+SW^2)*SA) : :  (barycentrics)

= On the Hutson-Parry circle, cubics K219, K237 and these lines: {2, 1340}, {115, 125}, {476, 1380}, {892, 6189}, {2039, 5996}, {2395, 5638}, {3413, 5466}, {6142, 6795}

= Tripolar centroid of X(3413)

= [ 3.673121323073036, 3.03523221804602, -0.155936895081187 ]

 

The squared-radius is:

R² = -2/9*K*OH^2*(SW+K)*(K^2-2*K* OH^2-2*SW*OH^2)/(K^4-4*OH^2*( SW+K)*(K^2-K*OH^2-SW*OH^2))

 

where K = sqrt(-3*S^2+SW^2)

 

ETC centers on the circle: X(13) and X(14)

 

Note: This circle is orthogonal to the orthocentroidal circle.

 

César Lozada