Circumcircles
By: Laura Lowe
Problem: The Nine-Point circle for any triangle passes through
the three mid-points of the sides, the three feet of the altitudes, and the
three mid-points of the segments from the respective vertices to orthocenter.
Construct the nine points, locate the center (N) and construct the nine point
circle. How is N related to G, H, C, or I for different shaped triangles?
To
begin I constructed the medial triangle ABC and its circumcircle. (Figure 1)
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Figure 1 |
Then
I constructed the orthic triangle and its circumcircle. (Figure 2)
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Figure 2 |
Last,
I constructed an acute triangle ABC.
I constructed H and the segments HA, HB, and HC and the midpoints of HA,
HB, and HC and connected the midpoints to form a triangle. Then I constructed the circumcircle of
this triangle. (Figure 3)
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Figure 3 |
It
appears upon visual examination that the circumcircles of the three secondary
triangles are the same. In other
words, they appear to have the same center (relative to the primary triangle)
and the same radii.
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Figure 1 |
Figure 2 |
Figure 3 |
When
I constructed all 9 points on the same triangle, the circle is called the 9
point circle. (Figure 4)
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Figure 4 |
By my
construction N is C. G, H, and I
are collinear as well. (Figure 5)
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Figure 5 |
These
relationships hold for all triangles, however, obtuse triangles do not have 9
point circles. (Figure 6)
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Figure 6 |
Why
is this? Try to construct a nine
point circle on an obtuse triangle and see what happens.