The Nine-Point Circle

Let ABC be any triangle and then let P,Q,R be the feet of the altitudes which intersect at H. Denote by X,Y,Z the midpoints of the sides and by J,K,L the midpoints of the segments AH, BH and CH. Then there is a circle that passes through all nine of the points P,Q,R,X,Y,Z,J,K,L. This is the Nine-point Circle.

To approach the proof that this circle exists, we start back at the beginning of Euclidean Geometry. Here I will present the basic facts that are needed - but I will leave all the proofs to you! Some of these results will be familiar and some will be new. It is not necessary to prove all these statements in order. Dive in somewhere and have fun!

Lemma 1 If points P and Q are on the same side of segment AB then the angles at P and Q are equal if and only if there is a single circle passing through all four points.

Lemma 2 The diameter of a circle determines (subtends) a right angle at the circumference. Conversely, if a segment AB is the hypotenuse of a right angle triangle then it the circle through the three vertices has the segment as a diameter.

Lemma 3 Consider a hexagon ABCDEF with the property that each pair of opposite edges is in fact the opposite ends of a rectangle. Then there is a single circle through all six vertices of the hexagon.

Lemma 4 If lines AB and CD are parallel and divide two lines through P into segments of length a ,b and x ,y respectively then a/b = x/y. So parallel lines divide all lines through P in the same ratio.

Lemma 5 In a triangle ABC (see figure below Step 1), let X,Y be midpoints of sides AB and AC. Then XY is parallel to BC and half as long.Step 1: In the figure below X,Y,Z are the midpoints of the sides and AP is the altitude at A. Then the triangles XPY and YZX are congruent and hence the angles XPY and YZX are equal. Therefore X,Y,Z and all the feet of the altitudes are on a single circle.

Step 2: In triangle ABC, the midpoints if the sides and the midpoints of AH, BH, CH - namely X ,J, Y, K, Z, L - are vertices of a hexagon with the properties of Lemma 3. (H is the point where the altitudes intersect.)

And that should complete your proof of the existence of the Nine-point Circle since the circle constructed in Step 1 must be the same as the circle constructed in Step 2 since they both contain X,Y and Z.