__Explorations__

By Brandie Thrasher

It is true that for an equilateral triangle, its
triangle of medians is also equilateral. What about an isosceles triangle? Will
its triangle of medians also be isosceles? Lets explore!

First, lets answer the question, what is a triangle of
medians?

Ok, well given this triangle

Construct the medians (a segment joining a midpoint of
a leg to an opposite vertices)

From the these segments, we create a triangle with
sides of equal length of these medians

The triangle shown with the yellow interior is the
triangle of medians for triangle ABC.

Now, given an isosceles triangle (JFI) letÕs construct
its triangle of medians ÆLHI

Since ÆJFI is isosceles having line segments JF and IF being
congruent to one another, there medians must also be equal, thus segments LI
and KJ are congruent to one another. In our triangle of medians, segment LH
represents segments segment JK, which is congruent to segment LI, thus our
triangle of medians is also isosceles with base IH.

What about other special triangles? Will a right
triangle generate a right triangle of medians?

Here is a right triangle with its median of triangle
constructed

As shown, ÆABC is in fact a right triangle, but the triangle of
medians (ÆDEC) is not a right triangle. It is in fact a scalene
triangle

Will this always be the result?

Here is a right isosceles triangle, and its triangle
of medians is isosceles, but not containing a right angle.

But, as our triangle expands where angle BAC
approaches 35.2û and angle 54.75û angle FGA of our
triangle of medians approaches and exceeds 90û, thus there is a
point in which a right triangle produces a right triangle of medians.