Math CentralQuandaries & Queries


Question from Patrick:

I'm trying to find a quick calculation to find the center of the circle that will touch the 3 points of an isosceles triangle.<br />
Known dimensions are length of the triangle base and the height from the top point to the middle of the base.<br />
Thanks in advance, <br />
Kind regards,<br />

Hi Patrick,

In my diagram the center $C$ of the circle is between the top point $P$ and the middle $M$ of the base.


Let the radius of the circle be $r$ and then $|PC| = |CR| = r.$ If $h$ is the height of the triangle then $|CM| = h-r.$ Let the distance from $M$ to $R$ be $b,$ which is half the length of the base of the triangle. Triangle $PMR$ is a right triangle. Use Pythagoras theorem for this triangle and solve for $r.$

Apply a similar argument to find $r$ if $M$ is between $P$ and $C.$


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