When doing circle geometry, what's the best way to go about proving -
1. 3 points are coolinear for instance - Prove that point C lies on AB
2. Proving something is a cyclic quad
3. Proving a line bisects an angle
1. Generally the easiest way is to prove that angle ABC is 180 degrees (angle sum of straight line). You do this by initially assuming they are not collinear, and finding angles ACX (for X somewhere else, depends on the question) and BCX and then summing.
2. Cyclic quad has a number of ways depending on question. Often the easiest way is to prove supplementary opposite angles or the associated exterior angle being the same as the opposite. Another way that works is if a line subtends the same angle at the two points because then you have the converse of angles standing on the same chord subtend equal angles at the circumference. This only works if they are on the same side. The final way (just going off my head so I might have missed some), is if the line subtends 90 degree angles on both sides which then gives the converse of an angle in a semicircle is 90 degrees.
3. Bisecting a line is just using all your circle geo properties to find the values of both subcomponents of the 'dissected' line and hence show they are equal.
