Assignment # 6

Fun with Medians!

We are to investigate the Medial Triangle - a triangle constructed out of the medians of any given triangle. To make life easier on us, let's construct them so they are connected.


 The Construction: Given a triangle MKH, if one was to construct a triangle using it's medians, a good construction can be done as follows:

  • Take an arbitrary segment (say LK), and from point L, construct a parallel line to IM (another median).
  • From point K, construct a parallel line to median HN.
  • The intersection of the two gives us the medial triangle.

1. Area

So now we have our triangle LKJ. What is the relationship to HMK? Let's look at some givens:

So now we can deduce that HLG is exactly one-fourth the area of HLK, making the medial triangle (since all else was similiar) three-fourths the area of our original triangle.

2. Isosceles-Right ...

Let's look at a right isosceles triangle ...


 A right isosceles triangle

We can actually make a square ADEF since triangle ADF is a right isosceles triangle as well.

By this, we get EC congruent to DC (corresponding sides), making EDC an isosceles triangle. But it's not a right one.

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