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# More Unusual Properties of Acute Triangles

**The
Nine-Point Circle** of Triangle ABC with orthocenter H passes through:

1. the midpoints L, M, and N of the three sides,

2. the feet of the altitudes D, E, and F to those sides, and

3. the points X, Y, and Z, which are the midpoints of the segments AH, BH, and CH, respectively.

1. **The Nine-Point Center U** lies on the Euler Line of Triangle ABC.

The Euler line is the line passing through

2. the orthocenter H,

3. the circumcenter CC, and

4. the centroid G of a triangle.

The
tangents to the Nine-Point Circle at the midpoints L, M, and N of the sides of
the triangle form a triangle, RST, that is similar to the orthic triangle (the
triangle DEF). In fact, the sides of this triangle are parallel to those of
triangle DEF.

For an extensive listing of 20 additional properties about the nine point
circle, see the following reference, specifically pages 53-56:

**MacKay, J. S. (1892). History of the Nine Point Circle. ***Proceedings of
the Edinburgh Mathematical Society, (11).* pages 19-61.

### Feuerbach's Theorem

The Nine-Point Circle of a triangle "touches"
the incircle and the three excircles.

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