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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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