CONCEPT 1 – Construct the circumscribed circle of a triangle (The Circumcircle)

The location of the circumcenter is sometimes investigated.  In a dynamic program like Geometer’s Sketchpad you are able to see that the circumcenter has great mobility such that it can be on, in or out of the triangle.

CONCEPT 2 – Construct the inscribed circle of a triangle (The Incircle)

The central construction theory to the inscribed circle is the angle bisector.  Its characteristics make this construction possible.  Often the characteristic of the angle bisector that we focus on is that it bisects the angles into two congruent angles but in this construction that is secondary to the fact that the angle bisector represents all points that are equidistant to the two sides of the angle. This can be proven using congruent triangles and the AAS congruency theorem (bisected angle, right angle, and common side).  The congruent triangles tell us that all points are equidistant to the sides of the angle.

The incenter never leaves the interior of the triangle.

CONCEPT 3 – Prove properties of angles for a quadrilateral inscribed in a circle.