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Comment: For construction shown in picture we assumed that centers of spheres are all on xy plane. Then the intersection of three spheres with xy plane were considered. Now in plane xy we used Descartes theorem for four mutually tangent circles. In this way the radii of circles which is tangent to three circles were found. The radius of small circle is 1.52 which is the lower bound of radius of sphere touching three other circle. Calculation due to Descartes theorem:

$(\frac 1a+\frac 1b+\frac 1c+\frac 1r)^2=2(\frac 1{a^2}+\frac 1{b^2}+ \frac 1{c^2}+\frac 1{r^2})$

Putting the following values:

$(a, b, c)=(15, 10, 7)$

We get $r=1.52$ and $r=25.2$

Update: I could find the radius of fourth circle using CAD software:

$r\approx 6$ and $r\approx 1.6$.

Comment: I could find the radius of fourth circle using CAD software:

$r\approx 7.585$ and $r\approx 24.765$ for $a=75$, $b=50$, $c=35$.

FRONT VIEW:

SIDE VIEW:

Comment: For construction shown in picture we assumed that centers of spheres are all on xy plane. Then the intersection of three spheres with xy plane were considered. Now in plane xy we used Descartes theorem for four mutually tangent circles. In this way the radii of circles which is tangent to three circles were found. The radius of small circle is 1.52 which is the lower bound of radius of sphere touching three other circle. Calculation due to Descartes theorem:

$(\frac 1a+\frac 1b+\frac 1c+\frac 1r)^2=2(\frac 1{a^2}+\frac 1{b^2}+ \frac 1{c^2}+\frac 1{r^2})$

Putting the following values:

$(a, b, c)=(15, 10, 7)$

We get $r=1.52$ and $r=25.2$

[![enter image description here][2]][2]

Comment: For construction shown in picture we assumed that centers of spheres are all on xy plane. Then the intersection of three spheres with xy plane were considered. Now in plane xy we used Descartes theorem for four mutually tangent circles. In this way the radii of circles which is tangent to three circles were found. The radius of small circle is 1.52 which is the lower bound of radius of sphere touching three other circle. Calculation due to Descartes theorem:

$(\frac 1a+\frac 1b+\frac 1c+\frac 1r)^2=2(\frac 1{a^2}+\frac 1{b^2}+ \frac 1{c^2}+\frac 1{r^2})$

Putting the following values:

$(a, b, c)=(15, 10, 7)$

We get $r=1.52$ and $r=25.2$

Update: I could find the radius of fourth circle using CAD software:

$r\approx 6$ and $r\approx 1.6$.

Comment: For construction shown in picture we assumed that centers of spheres are all on xy plane. Then the intersection of three spheres with xy plane were considered. Now in plane xy we used Descartes theorem for four mutually tangent circles. In this way the radii of circles which is tangent to three circles were found. The radius of small circle is 1.52 which is the lower bound of radius of sphere touching three other circle. For upper bound we may use the radius of big circle which is $25.2$, although I think there is no limit for that.

Calculation due to Descartes theorem:

$(\frac 1a+\frac 1b+\frac 1c+\frac 1r)^2=2(\frac 1{a^2}+\frac 1{b^2}+ \frac 1{c^2}+\frac 1{r^2})$

Putting the following values:

$(a, b, c)=(15, 10, 7)$

We get $r=1.52$ and $r=25.2$

The second picture shows the situation, the dark spheres are those their radius found:

Comment: For construction shown in picture we assumed that centers of spheres are all on xy plane. Then the intersection of three spheres with xy plane were considered. Now in plane xy we used Descartes theorem for four mutually tangent circles. In this way the radii of circles which is tangent to three circles were found. The radius of small circle is 1.52 which is the lower bound of radius of sphere touching three other circle. Calculation due to Descartes theorem:

$(\frac 1a+\frac 1b+\frac 1c+\frac 1r)^2=2(\frac 1{a^2}+\frac 1{b^2}+ \frac 1{c^2}+\frac 1{r^2})$

Putting the following values:

$(a, b, c)=(15, 10, 7)$

We get $r=1.52$ and $r=25.2$

[![enter image description here][2]][2]