By Frater Elijah

Construct a triangle and it's medians. Remember the medians connect a vertex and the midpoint of it's opposite side.

Construct a second triangle with the three sides having the lengths of the three medians from the first (above) triangle.

It seems that with an equilateral triangle, when constructing the medians and then rearranging these pieces into another triangle we will again get an equilateral triangle (but of different side length). Re-interpreting the above decimal interpretation into a different form, by the Pythagorean theorem we have the length of Cc to be

(Cc) ² + (Ac) ² = (AC) ²

ý (Cc)² + (.5) ² = 1²

ý (Cc) ² =1-.25 =.75

ý Cc = (.75)^.5 This is also the length of a median.

These triangles are most definitely similar, but what else can we see about them. We will refer to the above triangles as triangle I and II respectively. Now triangle I has the following statistics:

Side length: 1

Area =.5(base)(height) = .5(1)((.75)^.5) = (.75)^.5/2

Perimeter = 3

Triangle I is not congruent to Triangle II.

Now let us see the stats on triangle II.

Side length = (.75)^.5

Area = (.75)^.5/2 (height) What's the Height?

Perimeter = 3(.75)^.5

Let's find out using the Pythagorean theorem again.

Height² = [(.75)^.5]² - [(.75/2)^.5] ² = (.75) - (.75)/4 = (3/4) - (3/16)} =

(12/16) - (3/16) = (9/16)

So height = 9^.5/16^.5 = 3/4 = .75

So Area = (.75)(.75)^.5/2.

Does this make sense? Yes, because triangle II is smaller than triangle I, so the area will have to me smaller. Look at the ratio between the area relationships.

Triangle I has area (.75)^.5/2 = median side length/2

Triangle II has area (.75) (.75)^.5/2 = (.75) (area of triangle I)= (median side) ² (Area of I)

… let's x-plore one more case (see below).

The perimeters are in an obvious ratio. Why?

Basically because the construction of medians from an equilateral triangle gives rise to another equilateral triangle. So to get the perimeter, we are just multiplying one median by 3 (or side length in the starting case of side length 1).

Lets construct one more triangle from the medians of the above triangle and see what we can come up with.

Now reconstructing the medians gives rise to the below triangle

Now the side length is .75 so the perimeter is 3 (.75).

The triangle is equilateral .

The area = .75/2 (height). We need to find the damned height again.

(.75) ² =(.75/2)² + (h) ²

ý h² = (9/16) - (9/64) = (36/64) - (9/64) = 27/ 64

ý h = (27)^.5 /8

So the area = [(3/4)^3]^.5 (.75/2)

An interesting observation 27 and 64 are perfect cubes (roots 3 and 4 respectively), so the cube root of

27/64 is 3/4. This is our third iteration of this median construction also. Lets go back and see if we had

something from the second iteration. We had our height equal to h² = 9/16 which is composed of two

perfect squares 3 and 4. [we took the square root or the second root on the second iteration]. I'll say we

may be onto something here. It seems that we have taken (.75)^.5 = (3/4)^.5 which was the side length of

the triangle in our first median construction (triangle II) squared it then took then took the squareroot

(where n is the current stage we are at - the triangle number). In the third triangle we took (3/4), cubed it

(the current stage) and then took the square root (stage 3-1=2) to which gives rise to the side length for the

IV triangle. This is also used in the area formulation. Let's try this relationship and see if it works for the

next case. For triangle IV we have (3/4)^4 = 81/256. Then taking square roots yields (81/256)^.5 = 9/16.

Investigate this interesting relationship on your own and see what you can find.

The following pictures are with the median triangles inside of one another.

 When overlapped around a 30 degree point of rotation which corresponds to the initial creation pattern except it is layed out as to be visible, we see an interesting spiral.

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