Three Spheres and a Tetrahedron

문제

Given a tetrahedron $OABC$ with vertices $O$ , $A$ , $B$ and $C$ .

There is a sphere, $S1$ (red, center $Q1$ ), inscribed in the tetrahedron tangent to the inside of each face $OAB$ (gray), $OAC$ (brown), $OBC$ (magenta) and $ABC$ (cyan and black).

There is a second sphere, $S2$ (green, center $Q2$ ), tangent to the (extended) inside of $OAB$ , $OAC$ and $OBC$ and to the outside of $ABC$ . (There is actually such a sphere for each face, tangent to the outside of the face and the inside of the other extended faces).

There is a third larger sphere, $S3$ (blue, center $Q3$ ), which passes thru vertices $A$ , $B$ and $C$ and is tangent to each of $S1$ and $S2$ so the outside of the smaller spheres is tangent to the inside of the largest sphere (see Figure 1, below, for two different views. Triangle $ABC$ is cyan in the first picture and black in the second one for clarity):

Figure 1

The following figures give several views of the tetrahedron and spheres.

Figure 2 shows the view along $OA$ , which shows the two smaller spheres tangent to $OAB$ and $OAC$ (left). The view along $BC$ shows the two smaller spheres tangent to $OBC$ and tangent on opposite sides of $ABC$ (right):

Figure 2

Figure 3 shows $S3$ passing through $A$ , $B$ and $C$ and tangent to $S1$ and $S2$ . On the left, the view perpendicular to the plane of triangle $A$ , $B$ , $Q3$ shows $S3$ passing through $A$ and $B$ . In the center, the view perpendicular to the plane of triangle $A$ , $C$ , $Q3$ shows $S3$ passing through $A$ and $C$ . On the right, the view perpendicular to the plane of triangle $Q1$ , $Q2$ , $Q3$ (the centers of the three spheres) shows $S1$ and $S2$ tangent to the inside of $S3$ .

Figure 3

Write a program which takes as input the vertices $O$ , $A$ , $B$ and $C$ and computes the center and radius of the big sphere (which entails finding the other two spheres).

$O$ will be the origin $(0,0,0)$ . A will lie on the positive $x$ -axis $(Ax,0,0)$ , $B$ will be on the $xy$ −plane $(Bx,By,0)$ and $C$ will be in the first orthant $(Cx,Cy,Cz)$ . $Ax$ , $By$ and $Cz$ will be strictly positive and the remaining values will be non-negative.

입력

The input consists of a single line containing six double precision decimal values $Ax$ , $Bx$ , $By$ , $Cx$ , $Cy$ and $Cz$ in that order (as described above), ( $0 < Ax, By, Cz ≤ 10$ ) and ( $0 ≤ Bx, Cx, Cy ≤ 10$ ).

출력

The single line of output contains four decimal values to four decimal places: $center\_x$ , $center\_y$ , $center\_z$ and $radius$ of the big sphere.