Question 1170611
<br>
Since the side lengths are natural numbers, we are looking at triangles whose side lengths are a Pythagorean Triple.<br>
If the lengths of the legs are a and b with hypotenuse c, then the area is given by<br>
{{{A = (1/2)ab}}}<br>
and by<br>
{{{A = (1/2)(p)(r) = (1/2)(a+b+c)(r)}}}<br>
where r is the radius of the inscribed circle.<br>
Since we want the radius of the inscribed circle to be 4, we are looking for Pythagorean Triples in which<br>
{{{ab = 4(a+b+c)}}}
or
{{{ab/(a+b+c)=4}}}<br>
We can simply look at each Pythagorean Triple and see if the radius of the inscribed circle is 4:<br><pre>

   a,b,c       ab  a+b+c  r = ab/(a+b+c)
  -------------------------------
   3,4,5       12    12      1
   5,12,13     60    30      2
   7,24,25    168    56      3
   8,15,17    120    40      3
   9,40,41    360    90      4
  11,60,61    660   132      5</pre>
All larger Pythagorean Triples will make triangles in which the radius of the inscribed circle is greater than 4, so we don't need to look any further.<br>
We see from the list that there is one primitive Pythagorean Triple (9-40-41) that gives 4 for the radius of the inscribed circle.<br>
We can also see that we can get a radius of 4 by scaling up the 3-4-5 triangle by a factor of 4, or by scaling up the 5-12-13 triangle by a factor of 2.<br>
So there are exactly three right triangles with side lengths that are natural numbers in which the radius of the inscribed circle is 4:<br>
12,16,20
10,24,26
9,40,41<br>