To get at least 1 foot of space around the pool you have to shrink the triangle by some factor using the incenter as the center of dilation.
A
a
See solution.
B
b
Yes, the center remains in the same position.
Explanation: See solution.
Practice makes perfect
a
To achieve the largest possible pool, we want to find the triangle's incenter.
Finding the incenter
To locate the incenter, we have to determine at least two angle bisectors for each of the triangle's vertices. Let's start by drawing an angle bisector to ∠R First, we will draw an arc across QR and PR with an arbitrary radius like below.
Next, we draw two smaller arcs using the intersections of the first arc with QR and PR as our midpoints. Make sure you keep the compass settings the same for both of them.
The segment from P and through the intersection point of the two smaller arcs, is the angle bisector to ∠P.
We need one more angle bisector to find the incenter. Let's repeat the procedure for ∠Q.
Where the angle bisectors intersect, we find the triangles incenter. Using a perpendicular bisector, we find the radius of the circle centered at the incenter.
Since this point is equidistant from each side, the radius is the maximum possible without the pool leaving the triangle.
b
The center of the pool would remain the same, as we are keeping the same distance between each of the sides. The only point equidistant from each of the sides is the incenter. We would merely make the radius of the pool 1 foot less.
