:
For any triangle the following nine points lie on the same circle:
1) The three midpoints of the sides of the triangle,
2) The feet of the three altitudes of the triangle,
3) The midpoints of the three segments connecting vertices to the orthocenter.
This circle is called the Ninepoint Circle of the triangle.
Its center is the midpoint of the
segment between the orthocenter and the circumcenter and its radius is ½ the radius of the
circumcircle.
In the following discussion, the following three theorems are frequently applied :
Theorem 4.2.15, The Midpoint Connection Theorem:
If a line segment has as its endpoints
the midpoints of two sides of a triangle then the segment is contained in a line that is parallel to
the third side and the segment is onehalf the length of the third side.
Theorem (NIB) 4.6, The Hypotenuse Diameter Theorem:
For any right triangle, the circle
which has the hypotenuse as diameter contains the vertex with the right angle.
The Perpendicularity Statement:
If a line is perpendicular to one of two parallel lines, then it
is perpendicular to the other.
As shown before, in Neutral Geometry the Perpendicularity Statement is equivalent to the
Euclidean Parallel Postulate, so the Perpendicularity Statement is true in Euclidean Geometry.
Two Lemmas are used below and their proofs are left as exercises:
Lemma 1:
A parallelogram with at least one right angle is a rectangle.
Lemma 2: A rectangle is circumscribed by a circle and each diagonal of the rectangle is a
diameter of the circle.
Recall that, in Euclidean Geometry, Parallelism is Transitive
(by Theorem 4.2.9):
For given lines
l
,
m
,
and
n
,
if
l

m
and
m

n
then
l

n
.
Also, by the Perpendicularity Statement, for given lines
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 Spring '10
 Shirley
 Pythagorean Theorem, triangle, Elementary geometry, Ninepoint circle, Midpoint Connection Theorem

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