# 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!

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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

that is a pretty good start.

Hint:

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?)

work out the midpoint and calculate the distance to the centre.

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Thanks very much for your help, I got it now:) – Laila Apr 15 '13 at 0:41
@Laila no worries – Lost1 Apr 15 '13 at 10:41

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 $ON$

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