has lengths 
and 
with 
.
are 
and 
What is the value of ![\[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\]](http://latex.artofproblemsolving.com/9/3/8/938ecf6869b053c8335864493848733a3f242e2e.png)
Y by
Eight circles of radius 
can be placed tangent to side 
of 
such that the first circle is tangent to 
, subsequent circles are externally tangent to each other, and the last is tangent to 
. Similarly, 
circles of radius 
can also be placed along in this manner. The inradius of is 
, where 
and 
are relatively prime positive integers. Find 
.
can be placed tangent to side 
of 
such that the first circle is tangent to 
, subsequent circles are externally tangent to each other, and the last is tangent to 
. Similarly, 
circles of radius 
can also be placed along in this manner. The inradius of is 
, where 
and 
are relatively prime positive integers. Find 
.
,
.
,
is the inradius of 
,
and 
.
.
,
,
.
.![\[\frac{r-34}{r} = \frac{14 \cdot 34}{BC}, \quad \frac{r-1}{r} = \frac{4046 \cdot 1}{BC},\]](http://latex.artofproblemsolving.com/6/c/7/6c7507dae4ca4e6c24ae9d8061162f4c5d398472.png)
.
varies linearly, the distance 
between the centers of circles of two circles of radius 
and 
and one tangent to 
,
,
,
,
.
,
.![\[ \frac{r - 34}{476} = \frac{r - 1}{4046} \to r = \frac{192}{5} \to \boxed{197}.\]](http://latex.artofproblemsolving.com/1/6/0/1605624c09237b1a2c044a84eeca9eae3e9855b6.png)
![\[\frac{14\cdot 34}{BC}=\frac{r-34}{r}\qquad\text{and}\qquad\frac{4046}{BC}=\frac{r-1}{r},\]](http://latex.artofproblemsolving.com/9/3/a/93abd0a964e50da2e5c030821e0874769dd77018.png)
so![\[\frac{r-34}{r-1}=\frac{2}{17}\implies r=\frac{192}{5}\implies\boxed{197}.\]](http://latex.artofproblemsolving.com/b/3/c/b3cfb51302ed93230d6cbc524d20b81f953763b8.png)
be a right angle and it works. Makes calculations a little easier I guess?
be 
,
= 
which yields 
where we have used the fact that the ratio of heights in similar triangles is the same scale factor k as the ratio of sides.
