How do you find the largest sphere that you can draw in perspective?

Viewed from the top, it'd be this:

Added: on the frustum on the right, I've marked four points I think we know something about. We can unproject all eight corners of the frusum, and the centres of the near and far ends. So we know point 1, 3 and 4. We also know that point 2 is the same distance from 3 as 4 is from 3. So then we can compute the nearest point on the line 1 to 4 to point 2 in order to get the centre? But the actual math and code escapes me.

I want to draw models (which are approximately spherical and which I have a miniball bounding sphere for) as large as possible.

Update: I've tried to implement the incircle-on-two-planes approach as suggested by bobobobo and Nathan Reed :

function getFrustumsInsphere(viewport,invMvpMatrix) {
    var midX = viewport[0]+viewport[2]/2,
        midY = viewport[1]+viewport[3]/2,
        centre = unproject(midX,midY,null,null,viewport,invMvpMatrix),
        incircle = function(a,b) {
            var c = ray_ray_closest_point_3(a,b);
            a = a[1]; // far clip plane
            b = b[1]; // far clip plane
            c = c[1]; // camera
            var A = vec3_length(vec3_sub(b,c)),
                B = vec3_length(vec3_sub(a,c)),
                C = vec3_length(vec3_sub(a,b)),
                P = 1/(A+B+C),
                x = ((A*a[0])+(B*a[1])+(C*a[2]))*P,
                y = ((A*b[0])+(B*b[1])+(C*b[2]))*P,
                z = ((A*c[0])+(B*c[1])+(C*c[2]))*P;
            c = [x,y,z]; // now the centre of the incircle
            c.push(vec3_length(vec3_sub(centre[1],c))); // add its radius
            return c;
        },
        left = unproject(viewport[0],midY,null,null,viewport,invMvpMatrix),
        right = unproject(viewport[2],midY,null,null,viewport,invMvpMatrix),
        horiz = incircle(left,right),
        top = unproject(midX,viewport[1],null,null,viewport,invMvpMatrix),
        bottom = unproject(midX,viewport[3],null,null,viewport,invMvpMatrix),
        vert = incircle(top,bottom);
    return horiz[3]<vert[3]? horiz: vert;
}

I admit I'm winging it; I'm trying to adapt 2D code by extending it into 3 dimensions. It doesn't compute the insphere correctly; the centre-point of the sphere seems to be on the line between the camera and the top-left each time, and its too big (or too close). Is there any obvious mistakes in my code? Does the approach, if fixed, work?