Question

ABCD is a rectangle .prove that the center of a circle through A,B,C,Dis the point of internation of its diagonals

Answer 1

 Central Angle Theorem - Math Open Reference

and i quote The central anglesubtendedby two points on a circle is twice the inscribed anglesubtended by those points.


// diclaimer below diagram is for contradiction purpose only


// O is center of circle
now we know 
<ABC = 90 deg   
so <AOC = 180

 which means AOC is straight line joining A and C // hope that statement made sense :P


So AC is a straight line passing through O

one diagonal of rectangle passes through center.

similarly we can prove that second diagonal also passes through center 

so  both diagonal line segments pass through center of circle 

and when two different straight lines have a point in common they must intersect at that point

so diagonals of rectangle must intersect at center 

// hope it helps