Circumcenter

The circumcenter is the center of the circumcircle of a polygon. Only certain polygons can be circumscribed by a circle: all nondegenerate triangles have a circumcircle whose circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle. Quadrilaterals which have circumcircles are called cyclic quadrilaterals. Also, every regular polygon is cyclic.

Circumcircle2.PNG

Proof that the perpendicular bisectors of a triangle are concurrent

First Proof

We consider a nondegenerate triangle $\triangle ABC$. Since the triangle is nondegenerate, $AB$ and $BC$ lie on different lines and so their perpendicular bisectors are not parallel and thus intersect. Let $O$ be the intersection of these perpendicular bisectors. Since $O$ lies on the perpendicular bisector of $AB$, it is equidistant from $A$ and $B$; likewise, it is equidistant from $B$ and ${C}$. Hence $O$ is equidistant from $A$ and ${C}$; hence $O$ also lies on the perpendicular bisector of $AC$ (and is the circumcenter).

Second Proof

We start with a diagram:

Circumproof1.PNG

One of the most common techniques for proving the concurrency of lines is Ceva's Theorem. However, there aren't any cevians in the diagram which would be needed for a direct application of Ceva's Theorem. Thus, we look for a way to make some by drawing in helpful lines. Drawing in $DE, EF$ and $FD$ (i.e. the medial triangle of $ABC$) does the trick.

Circumproof2.PNG

By SAS Similarity $\triangle BFD\sim \triangle BAC$. Thus $\angle BFD = \angle BAC$ making $FD || AC$. Since $EO\perp AC$ and $AC\| FD, EO\perp FD$ making $EH$ an altitude of $DEF$. Likewise, $DG$ and $FI$ are also altitudes. Thus, the problem is reduced to proving that the altitudes of a triangle are concurrent. This can be done using Ceva's Theorem.

It is worth noting that the existance of the circumcenter is a much more fundamentally important theorem than it might seem, since it implies that three points determine a circle.

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