**The Nine-Point Circle
of a triangle passes through the midpoints of the sides, the feet
of the altitudes, and the three midpoints of the segments from
the respective vertices to the orthocenter. **

**We will look at two
properties of the nine-point circle.**

**1. The radius of the
nine-point circle is one-half the length of the radius of the
circumcircle. **

**2. The center of the
nine-point circle is the midpoint of the segment defined by the
orthocenter and the circumcenter.**

**Proof for #1:**

**Since Triangle DEF
is the medial triangle, it is similar to Triangle ABC. Similar
triangles have corresponding parts that are proportional (including
altitudes, angle bisectors, etc.) and we know that the segment
connecting the midpoints of any two sides of a triangle are parallel
to and 1/2 the length of the third side. Thereore, the distance
from C1 to C is twice the distance from D2 to F.**

**Proof for #2:**

**Since S and T are midpoints
of semgnets BH and CH, then ST is congruent to DE by the triangle
midsegment theorem. Therefore, triangles DEF and TSR are congruent
since they share the same circumcenter. Therefore, the triangles
are symmetric around the center C2 and their orthocenters will
also be symmetric. But since H is the orthocenter of triangle
RST and C1 is the orthocenter of triangle DEF, we know that C2
is the midpoint of HC1. **

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