So i have a hw problem that gives a picture of a broken dinner plate with a little bit of a rounded circular edge and then a jagged edge, and I am supposed to find the center of the plate with just a compass and straight edge.
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.