Three Spheres and a Tetrahedron

문제

Given a tetrahedron $OABCOABC$ with vertices $OO$, $AA$, $BB$ and $CC$.

There is a sphere, $S1S1$ (red, center $Q1Q1$), inscribed in the tetrahedron tangent to the inside of each face $OABOAB$ (gray), $OACOAC$ (brown), $OBCOBC$ (magenta) and $ABCABC$ (cyan and black).

There is a second sphere, $S2S2$ (green, center $Q2Q2$), tangent to the (extended) inside of $OABOAB$, $OACOAC$ and $OBCOBC$ and to the outside of $ABCABC$. (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, $S3S3$ (blue, center $Q3Q3$), which passes thru vertices $AA$, $BB$ and $CC$ and is tangent to each of $S1S1$ and $S2S2$ 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 $ABCABC$ 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 $OAOA$, which shows the two smaller spheres tangent to $OABOAB$ and $OACOAC$ (left). The view along $BCBC$ shows the two smaller spheres tangent to $OBCOBC$ and tangent on opposite sides of $ABCABC$ (right):

Figure 2

Figure 3 shows $S3S3$ passing through $AA$, $BB$ and $CC$ and tangent to $S1S1$ and $S2S2$. On the left, the view perpendicular to the plane of triangle $AA$, $BB$, $Q3Q3$ shows $S3S3$ passing through $AA$ and $BB$. In the center, the view perpendicular to the plane of triangle $AA$, $CC$, $Q3Q3$ shows $S3S3$ passing through $AA$ and $CC$. On the right, the view perpendicular to the plane of triangle $Q1Q1$, $Q2Q2$, $Q3Q3$ (the centers of the three spheres) shows $S1S1$ and $S2S2$ tangent to the inside of $S3S3$.

Figure 3

Write a program which takes as input the vertices $OO$, $AA$, $BB$ and $CC$ and computes the center and radius of the big sphere (which entails finding the other two spheres).

$OO$ will be the origin $(0,0,0)(0,0,0)$. A will lie on the positive $xx$-axis $(Ax,0,0)(Ax,0,0)$, $BB$ will be on the $xyxy$−plane $(Bx,By,0)(Bx,By,0)$ and $CC$ will be in the first orthant $(Cx,Cy,Cz)(Cx,Cy,Cz)$. $AxAx$, $ByBy$ and $CzCz$ 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 $AxAx$, $BxBx$, $ByBy$, $CxCx$, $CyCy$ and $CzCz$ in that order (as described above), (0 < Ax, By, Cz &le; 10) and (0 &le; Bx, Cx, Cy &le; 10).

출력

The single line of output contains four decimal values to four decimal places: $cente{r}_{x}center\_x$, $cente{r}_{y}center\_y$, $cente{r}_{z}center\_z$ and $radiusradius$ of the big sphere.

2 3 2 3 1 4

2.8563 0.8218 1.8305 2.1816

1 0 2 0 0 3

1.0000 1.2500 1.6667 2.0833