Janet Kaplan

Construction of the Nine-Point Circle

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Although not actually receiving its present day name until 1842, the Nine-Point Circle was first discovered in the early 1800s. Several mathematicians through the years have made discoveries concerning the Nine-Point Circle, but the primary credit is given to Euler and Feuerbach. As such, the Circle also bears their names.

The Nine-Point Circle is constructed by the composition of three sets of significant points of a triangle. These sets are a) the midpoints of the sides of the triangle, b) the feet of the altitudes drawn to the sides, and c) the midpoints of the segments from the respective vertices to the Orthocenter of the triangle (also known as Euler Points).

The Nine-Point Center U lies on the Euler Line of the triangle – the line passing through the orthocenter H, the circumcenter O, and the centroid G of the triangle.

This famous Circle shows up in many important theorems, one of which is Feuerbach’s theorem. It states that the Nine-Point Circle is tangent to the incircle of a triangle and to its three excircles, which are the circles outside of the triangle that are tangent to the three sides.

Here we will construct the Nine-Point Circle, demonstrating how all nine points lie on this very interesting Circle.

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1) Begin by using Geometer's Sketchpad to construct triangle ABC, as well as the midpoints of the sides.

2) Now construct the feet of the altitudes of triangle ABC. The point of intersection of these altitudes is known as the ORTHOCENTER.

3) Construct the third set of points. These are the midpoints of the segments from each vertex to the Orthocenter.

4) At this point you should be able to see the Circle. But how can we construct it? First we need to find the Nine-Point Center.

In the figure below you can see that L, M, Y and Z form a rectangle with an axis parallel to the base of triangle ABC. The intersection of the diagonals of the rectangle, U, is the center of the Nine-Point Circle!  We could just as easily construct a rectangle with points X, Z, L and N, and the intersection of these diagonals would also be U, the Nine-Point Center.

With U as the center, we can pick any point on the circle to construct the circle.

5) Finally, we can see that the Nine-Point Center is collinear with the Euler Line of triangle ABC. Below are listed the three points on the Euler Line and their definitions.

CENTROID (G)         Intersection of the medians (lines joining the vertices to the midpoints of the opposite sides)

CIRCUMCENTER (O)     Intersection of the perpendicular bisectors of the sides; also the center of a circumscribed circle

ORTHOCENTER (H)       Intersection of the altitudes (can be inside or outside)