## 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:

• HLIJ is a parallelogram, so G is the midpoint of LJ. That means that GK splits our medial triangle is half (area wise).
• KL splits our main triangle HMK in half as well, and GLK is inside HLK.
• I is the midpoint of HK, so LI splits HLK into two triangles of equal area.
• G splits HLI into two traingles of equal area, because it is a median as well.

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.

• So DG is congruent to GE.
• GC is congruent to GC.
• Angle DGF is congruent to angle EGF (since ADFE is a square).
• So ... by SAS, triangle GDC is congruent to triangle GEC ...

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