> 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 half-circles with > positive z-axis and perpendicular to the xy-plane. > Determine the parameters A, B, C of the plane A*x + > B*y + C*z = 1 , that tangents the three > half-circles. > > Best regards, > Avni
In the general case let EF = a , FD = b , DE = c , and
In this case the parameters A, B, C are as follows
A = (b^2-a^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 xy-plane of the respective touching points of the half-circles 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 xy-plane coincides with the tripolar line of the symmedian point.