I think those two theorem are two of the most complicated formulas I have ever seen; please prove it because I am not able to find proofs on the internet:
It is known that if the sides of an inscribed quadrilateral $ABCD$ (that is in the order $AB,BC,CD,DA$) have lengths $a,b,c,d$ respectively and $p$ is the semi perimeter of the quadrilatral, then:
Theorem 1: The length of diagonal $AC$ of the quadrilatral is equal to $$\sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}}\;.$$
Theorem 2: The radius of the circle that contains all the vertices of the quadrilateral is equal to $$\frac14\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(p-a)(p-b)(p-c)(p-d)}}\;.$$
By the way, has anyone seen those theorems in a geometry textbook with solution?
