This exploration attempts to address the following three questions:

- What is the Orthocenter of a triangle (and how do I know it is a single point)?
- What is the Nine-Point Circle of a triangle?
- How does the relationship between the center of the Nine-Point Circle and the Orthocenter affect the type of triangle formed? And other investigations!

*A definition: *The **orthocenter** of a triangle is
the common intersection of the three lines containing the altitudes
of a triangle.

Here are some pictures of different types of triangles and their orthocenters, E.

Notice that the orthocenter does not always lie inside the triangle. This fact should not be a surprise, since we know that the altitudes of a right triangle can coincide with the sides of the triangle and obtuse triangles have some altitudes that lie outside the triangle. In triangle ABC above, two of the altitudes, AF and CB, lie outside the triangle (except for an endpoint, points A and C.) Noticing this fact now is important for later, when we consider how the Nine-Point Circle behaves for different types of triangles.

Of course, the definition of orthocenter begs a question...how
do we *know* the three lines containing the altitudes always
intersect at one point? **Click
here** for one way to prove that the lines containing the
altitudes of a triangle are concurrent.

*A definition:* The Nine-Point Circle for any triangle
passes through the three midpoints of the
sides of the triangle, the three feet
of the altitudes, and the three midpoints
of the segments from the vertices of the triangle to the
orthocenter. Note that this definition is NOT the only definition,
and it is perhaps not the most elegant. For an alternative definition,
see the **investigations** below.

Here's a picture of a Nine-Point Circle in an acute, scalene triangle:

*How to construct the Nine-Point Circle:* Obviously, first
you need to construct a triangle. **ADVICE: Start your construction
using lines for the sides of the triangle, not segments. (That
is, draw line AB, line AC, line BC. You can then draw segments
and hide the lines. **If you follow this advice, later on points
won't disappear from your Nine-Point Circle.

Then, with a little knowledge of The Geometer's Sketchpad,
constructing midpoints of the sides (**X**,
**Y**, and **Z**
above), and the altitudes of the triangle (with feet **D**, **F**,
and **G** above) is not terribly
difficult. Even the construction of the midpoints of the segments
AE, BE, and CE is not hard (points **U**,
**V**, and **W**
above.)

But, once you have the nine points, how do you create the circle?
Of course, you need to find the circle's center (**N**
above.) And this point **N** must
be equidistant from all of those nine points...so, you might consider
the locus of points equidistant from the endpoints of a segment...**Click here** for one way to find
**N**. Once you locate **N**, of course, you can draw the circle
by selecting the center and a point on the circle.

**Download this GSP file**
of a Nine-Point Circle and drag the vertices of the triangle to
explore these questions:

- The center of the Nine-Point Circle (
**N**) has a special relationship to some triangles located inside triangle ABC. What relationships can you find between**N**and some of these other special triangles (i.e., the medial triangle or orthic triangle)? - What happens to the Nine-Point Circle when the triangle is right?
- Can the orthocenter (
**E**) and center of the Nine-Point Circle (**N**) ever coincide? If so, what happens to the triangle when they do? If not, why not? - Determine what types of triangles are formed and how many
of the nine points are distinct when

-the orthocenter (**E**) is outside but the center of the Nine Point Circle (**N**) is inside the triangle;

-**E**coincides with a vertex of the triangle;

-**N**coincides with a vertex of the triangle;

-**E**and**N**are both outside the triangle

-**E**and**N**are both inside the triangle;

-**N**is outside but**E**is inside.

If any of the above are not possible, try to explain why not.

Happy Exploring...for this author's results, **click here**.

**Finding the center of a circle
given three (or more) points on a circle**--

Draw two chords and find the perpendicular bisectors of each chord. All points on a perpendicular bisector of a segment are equidistant from the endpoints of that segment. So, the point where the perpendicular bisectors meet will be equidistant from all three points and will be the center of the circle passing through all three points.

**Back** to Where I Was...