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## #1 2015-09-05 19:48:21

Bob
Registered: 2010-06-20
Posts: 9,863

### 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

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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