Geometry – Properties of Centers of Circumcircle, Incircle, and Nine Point Circle

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About 2 hours ago I came up with a very cool feature of Euclidean geometry, but it's much more difficult than I can handle, any help would be appreciated.

If we draw the circumcircle, the incircle and the nine point circle, then the circle passing through the centers of these three circles is congruent with the nine point circle if and only if it is tangent to the circumcircle.

Is this property already known?

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If the answer is yes, please include references that you can remember.

How do we prove that anyway?

Best Answer

This is actually known and follows from the statements:

  1. The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle (see).
  2. If a circle passes through the center of a circumcircle and tangents to it, then its radius is twice the radius of the circumcircle.
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