The Relationship Between the Incircle, Nine-Point Circle, and Three Excircles of a Triangle
Is there a relationship between the incircle, the nine-point circle, and the three excircles of a triangle? Lets sketch it and see.
Construct any triangle ABC. Use lines to connect A, B, and C instead of segments. Then construct segments connecting A, B, and C. This construction will require both. The incenter, I, is the intersection of the interior angle bisectors of triangle ABC. Construct I. Highlight one side of the triangle and point I. Construct a perpendicular line. Create the intersection of the perpendicular line and the side you highlighted. Highlight, in order, point I and the intersection you just created. Construct a circle by center and point. This is the incircle. Hide the perpendicular line, the angle bisectors, and the intersection point. Your sketch should look like this:
Now lets create the three excircles. Place a point on each line extension. Construct the exterior angle bisectors for a particular side of the triangle. For example, choose side AC. Construct the exterior angle bisector of the angles with vertex A and vertex C. Construct the intersection of the two bisectors. This is one excenter. Highlight the excenter and the line through AC. Construct a perpendicular line. Construct the intersection of the perpendicular line and AC. Highlight, in order, the excenter and the intersection point. Construct a circle by center and point. This is one excircle. Repeat this process for sides AB and BC. Hide the angle bisectors, perpendicular lines, intersections, and the points on the line extensions. Your sketch should look like this:
Now lets construct the nine-point circle. The nine points consist of the feet of the three altitudes, the midpoints of the segments connecting each vertex to the orthocenter, and the midpoints of each side. The center of the nine-point circle is the midpoint of the Euler line, which connects the orthocenter and the circumcenter.
Let begin by constructing the orthocenter, H. Construct the three altitudes. Construct the intersection of each altitude with the side of the triangle. The intersection of the altitudes is the orthocenter. Mark it H. Construct a segment from each vertex to H. Construct the midpoints of each of these three segments. Hide the altitudes and the segments from the vertices to H. Construct the midpoint of each side of the triangle. Construct the perpendicular to each side through each midpoint. The intersection of these perpendiculars is the circumcenter. Mark it C. Hide the perpendicular lines. Construct the segment connecting H and C. Construct its midpoint. Mark it M. This is the center of the nine-point circle. Hide point H, point C, and the segment connecting them. Highlight, in order, point M and one of the nine points on the nine-point circle. Construct a circle by center and point. Your sketch should look like this:
So what is the relationship? The three excircles and the incircle are all tangent to the nine-point circle. The incircle is inside of the nine-point circle. The three excircles are outside of the nine-point circle. Is this always the case? Click HERE for a GSP sketch that you can move.
Yes, this is always the case.