<![CDATA[Math Is Fun Forum / The Circumcircle]]> 2015-09-06T07:48:21Z FluxBB http://www.mathisfunforum.com/viewtopic.php?id=22507 <![CDATA[The Circumcircle]]> In any triangle, there is a unique circle that goes through the three vertices.  The centre is called the circumcentre. Find the midpoints D, E and F of the sides.

Construct the perpendicular bisectors of the sides (the line that goes through the midpoint and is at 90 to the side).

Call the point where the bisector from D and from E meet, point O.

In triangles OBD and OCD, one side is common (OD), BD = DC, and ODB = ODC = 90.  So the triangles are congruent (SAS).

So OB = OC.

Similarly OC = OA.

So in triangles OAF and OBF, OA = OB, OF is common and AF = BF, so the triangles are congruent (SSS).  So angle OAF = OBF = 90.

So the third bisector goes through O as well.

A circle, centre O, with radius OA will also go through B and C.

Bob

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http://www.mathisfunforum.com/profile.php?id=67694 2015-09-06T07:48:21Z http://www.mathisfunforum.com/viewtopic.php?pid=367140#p367140