Lots of questions.

First, to show that AB and BC are perpendicular, calculate the slope of the line between A and B. Call this m1. Then calculate the slope between B and C. Call this m2. If the two lines are perpendicular then .

For the second question. AC is a diameter, so to find the center of the circle you need the midpoint of AC. You can find this by the formulas:

and

The radius will be half the diameter, which is AC. So find the length of AC and divide it by 2. Call this r.

The equation of the circle of which AC is a diameter is

Finally, to show that B lies on this circle, all you need to do is plug the coordinates of B into the circle equation you just wrote down. If the LHS and RHS match, then B is on the circle.

-Dan