# Diameter of a circle with 3 coordinates

The question is: A circle has the points $A=(6,-1)$ $B=(10,-3)$ and $C=(-2,-9)$ on its circumference. A diameter of the circle is drawn which is parallel to BC. How far apart are the two parallel lines?

I managed to get the center of the circle $(5,-8)$, however I am now stuck. Thanks for you help!

• What is the geometric definition of a circle? (What is special about all the points on the circle?) – Sammy Black Apr 14 '13 at 21:58
• Or if you draw a normal from the center of the circle to BC, where do you think it will hit BC? (Hint: Almost no calculation needed. Draw a picture.) – Harald Hanche-Olsen Apr 14 '13 at 22:02

the line through $BC$ must hit the circle at $BC$. the distance between the mid-point of BC and the centre must be the same as the shortest distance from the BC and its parallel diameter (draw a picture and convince yourself, and why is this true?)
If center of circle is $O(5,-8)$ You need to find the straight line $n$ that is perpendicular to line $BC$ wich passes to $O$ then you find the point $N=n\cap BC$ finally you look for distance $ON$