Fuss-ional Bicentric Integral Quadrilateral

Some equations for a bicentric quadrilateral with sides $a$ , $b$ , $c$ , and $d$ include:

$a + c = b + d$

$A = \sqrt{abcd}$

$P = a + b + c + d$

$r = \cfrac{\sqrt{abcd}}{a + c} = \cfrac{2A}{P}$

$R = \cfrac{1}{4}\sqrt{\cfrac{(ab + cd)(ac + bd)(ad + bc)}{abcd}} = \cfrac{\sqrt{(ab + cd)(ac + bd)(ad + bc)}}{4A}$

$d = x = \sqrt{R^2 + r^2 - r\sqrt{4R^2 + r^2}}$ (Fuss' Theorem)

The following computer program brute forces every possibility for $0 < a < 100$ , $a \leq b < 100$ , $b \leq c < 100$ , $c \leq d < 100$ and finds that the only set with integer solutions is $a = 42$ , $b = 42$ , $c = 56$ , $d = 56$ , $A = 2352$ , $P = 196$ , $r = 24$ , $R = 35$ , $x = 5$ , and $r + R + A + P = \boxed{2607}$ .

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import math
for a in range(1, 100):
    for b in range(a, 100):
        for c in range(b, 100):
            d = a + c - b
            if d >= c:
                A = math.sqrt(a * b * c * d)
                P = a + b + c + d
                r = 2.0 * A / P
                R = math.sqrt((a * b + c * d) * (a * c + b * d) * (
                    a * d + b * c)) / (4 * A)
                if R ** 2 + r ** 2 - r * math.sqrt(4 * R ** 2 + r ** 2) > 0:
                    x = math.sqrt(R ** 2 + r ** 2 - r * math.sqrt(
                        4 * R ** 2 + r ** 2))
                    if (A - int(A)) == 0 and (r - int(r)) == 0 and (
                        R - int(R)) == 0 and (x - int(x)) == 0:
                        print(a, b, c, d, A, P, r, R, x, r + R + A + P)

Using the smallest right triangle available, we want the r to be divisible by both legs while also having an even diameter for the circumcircle, that equate having an even hypothenuse, so that the R is an integer.
r = (product of smallest legs) × (evening factor)
= 3 × 4 × 2
= 24

R = 5 × (3 + 4)
= 35

A = [ 2 × 3 × (3 + 4) ] × [ 2 × 4 × (3 + 4) ]
= 2352

P = 2 × { [ 2 × 3 × (3 + 4) ] + [ 2 × 4 × (3 + 4) ] }
= 196

Answer
= 24 + 35 + 2352 + 196
= 2607