Triangles and the
Medians

Given a triangle ABC and its medians AD, BE, and CF, construct
a triangle with sides the same length as the three medians.

a. Show that the triangle is unique. Use GSP to have a linked
construction. That is, if triangle ABC is changed, then the corresonding
triangle of medians changes also.

b. Show that the area of the triangle of medians is three-fourths
of the area of the given triangle.

c. Show that the ratio of the perimeter of the triangle of
medians to the perimeter of the given triangle is between .75
and 1.0.

See** EMT
668 Assignment 6, Items 1, 2, & 3**.

Prove that a median of a triangle divides the triangle
into two equal areas.
Hint:

Prove that the three medians of a triangle divide the triangle
into six equal areas.

Prove that the three medians
of a triangle are concurrent.

Given three segments
that are the medians of a triangle, show a construction for the
triangle.

Prove that the area of a triangle is
given by
where u, v, and w are lengths of the triangle's medians.

(Problem **4651, **Proposed by Khiem V.
(Thomas) Ngo, Falls Church, VA., in School Science and Mathematics,
Volume 98, Number 2, February 1998, p. 110.)

**Useful Result: **The triangle formed by the medians of
a given triangle will have an area three-fourths the area of the
given triangle.

If ABC is a triangle with medians of lengths u, v, and w, and
CGF is a triangle with sides the same length as these medians
then the ratio of the area of triangle ABC to the area of triangle
CGF is 4 to 3. The triangle CGF can be constructed by making segment
FG parallel and congruent to median BE and segment CG parallel
and congruent to median AD. The ratio of the areas can be proved
in several ways, but suffice to see triangle AFC is half the area
of ABC, triangle CHF is half the area of triangle CGF, and triangle
AHF is one fourth the area of triangle AFC. Therefore the ratio
of the areas of triangle AFC to FHC is 4 to 3 and so the ratio
of the area of triangle ABC to CGF is 4 to 3.

**Next step.**

Use **Heron's formula**
for the area of the triangle with sides of length u, v, and w.
The area of the triangle with medians of length u, v, and w is

** **where .
Substituting and simplifying leads to the result.

The **Details**
(if you need them).

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