Medians of triangles-Explorations

By Doug Griffin

Construct a triangle and its 3 medians. Construct a second triangle whose sides are the lengths of these 3 medians. What relationships exist between the two triangles? Are they congruent or similar? What about their perimeters and areas? Make some conjectures and prove them.

Click Here to show a triangle with its 3 medians shown and the triangle of medians constructed.

Now lets take a look at triangles which are isosceles triangles to see if we see anything.

Using our trimedians gsp program, we produce the triangle of medians gcf above. Given that triangle abc is isosceles (b is constructed anywhere along the perpendicular bisector of ac) we see first that gf = cf because a circle with center f passes through both c and g. Therefore triangle cgf is certainly isosceles. We could arrive at the same conclusion by proving triangle acd congruent to triangle caf (sas) and therefore cf = ad. ad=fg by construction, therefore cf=fg and triangle cfg is isosceles.

 

In the case shown above where triangle abc is equilteral we posit the conjecture that triangle gcf must also be equilateral. Since we proved that cf=fg above ( an equilateral triangle also being isosceles of course) it remains to be shown that cg also equals either of the two other segments.

A circle constructed with center point g and radius gf passes through c and therefore cg = gf by definition and triangle gcf is equilateral.

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