The 9 Point Circle

The First Three Points

Consider any triangle, ABC, and construct its altitudes which meet the sides at points D, E and F.  The three altitudes meet in a common point, H, which is called the orthocenter of the triangle:

Three More Points

Now find the midpoints of segments AH, BH and CH:

The Last Three Points

Next locate the midpoints of the sides of the triangle:

Then the nine points, D, E, F, K, L, M, A', B', C' all lie on a circle.

Proof:  Page 1

C'B' is a midsegment of triangle ABC, so parallel to BC and half its length:

Proof:  Page 2

In triangle HBC, LM is a midsegment, so it is also parallel to BC and half its length, and therefore parallel and equal to C'B'.

Proof:  Page 3

In triangle ABH, C'L is a midsegment, so it is parallel to AH, and therefore to AD:

But AD is an altitude of triangle ABC, so it is perpendicular to BC, and therefore C'L is perpendicular to LM and C'L.

Proof:  Page 4

Similarly, B'M is a midsegment of triangle AHC, so it too is parallel to AH and therefore AD:

Proof:  Page 5

Therefore, the red quadrilateral C'B'ML is a rectangle, and the intersection of its diagonals is the center of a circle which contains its vertices.

Proof:  Page 6

By similar reasoning, we can conclude that quadrilateral KLA'B' is also a rectangle, and its diagonals are diameters of the same circle:

Proof:  Page 7

Likewise, quadrilateral KC'A'M (green) is a rectangle, and its diagonals are also diameters of the same circle:

Proof:  Page 8

Thus the nine points K, E, B', M, A', D, L, C', F are vertices of a cyclic pentagon, and the circle containing them is called the "nine point circle of triangle ABC":