
Re: Three halfcircles
Posted:
Sep 6, 2011 3:06 PM


> Let DEF be a rectangular triangle such that D=[0 0 > 0], E=[3 0 0] and F=[0 4 0]. On the sides DE, DF and > EF are constructed respective halfcircles with > positive zaxis and perpendicular to the xyplane. > Determine the parameters A, B, C of the plane A*x + > B*y + C*z = 1 , that tangents the three > halfcircles. > > Best regards, > Avni
In the general case let EF = a , FD = b , DE = c , and D=[0 0 0], E=[c 0 0], F=b*[ca , sqrt(1ca^2) , 0] ,
where ca=(b^2+c^2a^2)/(2*b*c) .
In this case the parameters A, B, C are as follows
A = (b^2a^2)/(c*b^2) , B = (c^2*(a^2+b^2)(a^4+b^4))/(4*b^2*c*S) , C = 2*a/(b*c) ,
where S is the area of triangle DEF.
Let D1, E1, F1 be the projections on the xyplane of the respective touching points of the halfcircles with the tangent plane. The lines DD1, EE1, FF1 are concurrent at the symmedian point X(6) of triangle DEF. Furthermore, the intersection of the tangent plane with the xyplane coincides with the tripolar line of the symmedian point.
Best regards, Avni

