Circle puzzle involving 5 nodes A circle has points A, B, C, D, E positioned on the edge such that the straight line distance AB is 1922, BC is 1798, CD is 1798, DE is 2162 and EA is $x$. BE is the diameter of the circle.

What is the value of $x$?

Am I right in thinking to solve this I make simultaneous equations for the radius and then solve x that way?

Exchanging the lengths $CD$ and $DE$ to get a trapezium $BCD'E$, the following two equations hold for the diameter $s$ and the trapezium diagonal $d$.


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*Ptolemy's theorem: $1798^2+2162s=d^2$

*Thales's theorem (followed by Pythagoras's): $s^2=d^2+1798^2$


Thus $1798^2+2162s=s^2-1798^2$, or $s^2-2162s-2\cdot1798^2=0$. Solving gives the valid solution as $s=3844$, whence $x=\sqrt{3844^2-1922^2}=3329.00165\dots$
