**The line created by the
orthocenter (H), centroid
(G), and the circumcenter (C) of a triangle is reffered to as Euler's
Line. The endpoints of this line are always H and C, with G found
between these two points. This line also contains relationships
to a point called the incenter
(I). To investigate
the relationships present in the Euler Line and the incenter,
we need to examine various different types of triangles.**

**What do you think the relationship
between HG and HC is?**

**Does anything unusual happen
when the triangle is a right triangle? Construct a right triangle
and observe the relationships.**

**For all right triangles,
the centroid is located on a midpoint of the hypotenuse and the
orthocenter is located on the vertex across from the hypotenuse.
Why do you think this occurs?**

**Let's investigate the difference
between an acute and an obtuse triangle.**

**Did you notice that the
Ueler Line is contained inside the acute triangles and outside
the obtuse triangles? Which points remain inside the triangle
no matter what and which points can be both inside and outside
the triangle? When and where do those points exit the triangle?
To see the answers,
click here.**

**We will now explore the
relationships in an isosceles triangle. The picture below shows
that when the triangle is isosceles, the Euler's Line is located
along the perpendicular bisector of the non-equal side. Itis also
interesting to note that the incenter falls on the Euler's Line
in this triangle, while it did not in any of the other triangles
we explored. Why do you think that is?**

**Another interesting relationship
to be explored is the relationship of this line when the triangle
is an equilateral triangle. Manipulate the sides of the triangle
and examine what happens as the lengths of the side approach equality.
As the lengths of the sides of triangle ABC approach equality,
H, G, and C converge to one point, which converges also to the
incenter I.**