Assignment 8

A Proof of Congruent Circumcircles

By

Ken Montgomery

Triangle *ABC* is constructed
(Figure 1) using *The Geometer’s Sketchpad*©.

Figure 1:

A tool for construction of the orthocenter (Assignment 5) is used to construct the
orthocenter, H. Segments
,
, and
are constructed in red (Figure 2).

Figure 2:
with orthocenter, H

The circumcenter, O is constructed by a tool
(Assignment 5), and given in Figure 3 with segments connecting it to the three
vertices of the triangle.

Figure 3:
with orthocenter, H and circumcenter, O

The circumcircle for
is then constructed in purple (Figure 4).

Figure 4:
with circumcircle in purple

We then use our tools to construct the
circumcenter, O’ and circumcircle (red) of
(Figure 5).

Figure 5:
with circumcenter, O’ and circumcircle in red

At this point we hypothesize that the
circumcircles for
,
,
and
are mutually congruent. We prove this as a theorem.

**Theorem**: The
circumcircles for
,
,
and
are congruent

**Proof**:

We first construct the segments,
and
(Figure 6).

Figure 6:

Now since AC is a shared side of
and
, then by definition, both circumcenters, O and O’ lie
on the perpendicular bisector of AC (Figure 7).

Figure 7: O and O’ both lie on the
perpendicular bisector of

Quadrilateral COAO’ is formed by segments CO, OA, AO’ and
O’C. The diagonals of the quadrilateral are AC and OO’. Since
, quadrilateral COAO’ is a rhombus and by definition
we have
. Since OA is the radius of the circumcircle for
and O’A is the radius of circumcircle for
, the circumcircles are congruent and without loss of
generality, the circumcircles for
,
,
and
are also congruent (Figure 8).Ò

Figure 8: The circumcircles for
,
,
and
are congruent

Download Assign08KM.gsp
to explore this problem.

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