Spoiler alert!

Say the diameter is a segment from A=(a,b) to B=(c,d). A point C=(x,y) on the circle forms a *right* triangle with those two points (Thales!), so the vectors AC and BC are orthogonal, so their slopes are negative reciprocals. That is,

(y-b)/(x-a)=-(x-c)/(y-d).

— Alex Kontorovich (@AlexKontorovich) November 23, 2020