Circumcircle and Incircle

Circumcentre

The circumcircle is a triangle’s circumscribed circle, i.e., the unique circle that passes through each of the triangles three vertices. The center of the circumcircle is called the circumcentre, and the circle’s radius is called the circumradius. A triangle’s three perpendicular bisectors , , and meet at .

How to produce a perpendicular bisector

A perpendicular bisector of a line segment is a line segment perpendicular to and passing through the midpoint of (left figure). The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centred at and with radius and connecting their two intersections. This line segment crosses at the midpoint of (middle figure). If the midpoint is known, then the perpendicular bisector can be constructed by drawing a small auxiliary circle around , then drawing an arc from each endpoint that crosses the line at the farthest intersection of the circle with the line (i.e., arcs with radii and respectively). Connecting the intersections of the arcs then gives the perpendicular bisector (right figure). Note that if the classical construction requirement that compasses be collapsible is dropped, then the auxiliary circle can be omitted and the rigid compass can be used to immediately draw the two arcs using any radius larger that half the length of .

The perpendicular bisectors of a triangle are lines passing through the midpoint of each side which are perpendicular to the given side. A triangle’s three perpendicular bisectors meet at a point known as the circumcentre , which is also the centre of the triangle’s circumcircle.

Incircle

An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each of the polygon’s sides. The centre of the incircle is called the incentre, and the radius of the circle is called the inradius.

While an incircle does not necessarily exist for arbitrary polygons, it exists and is moreover unique for triangles, regular polygons, and some other polygons but for our course just need to know about the case with triangles.

The incentre is the point of concurrence (where they cross) of the triangle’s angle bisectors.

The (interior) bisector of an angle, also called the internal angle bisector , is the line or line segment that divides the angle into two equal parts.

The angle bisectors meet at the incentre .

How to bisect an angle

Given. An angle to bisect. For this example, angle ABC.

Step 1. Draw an arc that is centered at the vertex of the angle. This arc can have a radius of any length. However, it must intersect both sides of the angle. We will call these intersection points P and Q This provides a point on each line that is an equal distance from the vertex of the angle.

Step 2. Draw two more arcs. The first arc must be centered on one of the two points P or Q. It can have any length radius. The second arc must be centered on whichever point (P or Q) you did NOT choose for the first arc. The radius for the second arc MUST be the same as the first arc. Make sure you make the arcs long enough so that these two arcs intersect in at least one point. We will call this intersection point X. Every intersection point between these arcs (there can be at most 2) will lie on the angle bisector.

Step 3. Draw a line that contains both the vertex and X. Since the intersection points and the vertex all lie on the angle bisector, we know that the line which passes through these points must be the angle bisector.

 

To remember which construction to use I think of the mnemonic:

Artificial  Intelligence or Living Creature

Bisect:

           Angles for Incircle            or         Lines (sides) for Circumcentre.

Or better still reason it out by recalling that one of our theorems says that if you draw a tangent to a circle, then the radius to the point of tangency makes a right angle with the tangent. Each side of the triangle around an incircle is a tangent to that circle.