In an unbroken circle,

if you joint the centre to any 2 points on the circumference,

all 3 points then form an isosceles triangle.

An isosceles triangle is formed of 2 identical back-to-back right-angled triangles, which means that if you bisect the base of an isosceles triangle,

then join the base midpoint to the 3rd vertex, you will have formed identical right-angled triangles inside the isosceles triangle.

Going in reverse, if you draw a chord across the round part of the broken plate, then the isosceles triangle base is the line joining the 2 points on the circumference (the chord).

Hence if you bisect this chord and draw the perpendicular bisector, the circle centre lies on this line.

Draw a 2nd chord, bisect it, draw a 2nd perpendicular bisector.

The centre also lies on this line, hence the circle centre is the point of intersection of the perpendicular bisectors of the chords.