# Nine Point Circle: What Is This About?

A Mathematical Droodle

What if applet does not run? |

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Copyright © 1996-2018 Alexander BogomolnyThe applet purports to suggest a proof for the existence of the 9-point circle. The proof was brought to my attention by Hubert Shutrick. Here is the statement of existence:

In ΔABC, the midpoints of the sides M_{A}, M_{B}, M_{C}, the feet H_{A}, H_{B}, H_{C} of the altitudes, and the midpoints A_{H}, B_{H}, C_{H} of the segments connecting the orthocenter with the vertices, lie on a circle, known as the *9 point circle*.

The proof is illustrated by the applet:

First of all, M_{B}M_{C} is the midline in ΔABC so that it's parallel to BC and equals half of the latter. The same holds of B_{H}C_{H} which is a midline in ΔHBC. It follows that M_{B}M_{C}B_{H}C_{H} is a parallelogram. But more is true. M_{B}C_{H} is a midline in ΔAHC. In particular, this implies that M_{B}C_{H} is orthogonal to BC (as it is parallel to AH.) Hence the quadrilateral M_{B}M_{C}B_{H}C_{H} is a rectangle.

Observe that the diagonals of a rectangle (of a parallelogram in fact) cross at their midpoints. Let N be the center of the rectangle M_{B}M_{C}B_{H}C_{H}.

Rectangles M_{C}M_{A}C_{H}A_{H} and M_{A}M_{B}A_{H}B_{H} are obtained in a similar way. Between them, the three rectangles share three diagonals: M_{A}A_{H}, M_{B}B_{H}, M_{C}C_{H} and therefore have a common center. This shows that 6 points - M_{A}, M_{B}, M_{C}, A_{H}, B_{H}, C_{H} lie on a circle with center N.

Furthermore, in ΔA_{H}H_{A}M_{A} the angle at H_{A} is right whereas the hypotenuse A_{H}M_{A} serves as a diameter of the just found circle. It follows that H_{A} also lies on that circle, and similar argument applies to the feet of the remaining altitudes, H_{B} and H_{C}.

(**Note**: Hubert later suggested a shortcut. Once it was established that M_{B}M_{C}B_{H}C_{H} is cyclic, then the right angles give that H_{C} and H_{B} also lie on the circle. The similar circle defined by M_{C}M_{A}C_{H}A_{H}H_{C}H_{A} has the points M_{C}, C_{H}, H_{C} in common with the previous circle so they are the same. Another simple argument can be found elsewhere at the site.)

### Nine Point Circle

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Copyright © 1996-2018 Alexander Bogomolny