Convex quadrilateral ABCD has an incircle (that is, there is a circle that is tangent to every side). Let a, b, c, and d be the lengths of the tangents from the vertices to the circle, as shown. Prove that
\([ABCD]^2 = (a + b + c + d)(abc + abd + acd + bcd)\).
I haven't been able to do much but I was able to say that \(r^2(a+b+c+d) = abc + abd + acd + bcd\).