Triangle Proof

by Kathy McDonald

section 3.1 #7

Prove: When dividing each side of an equilateral triangle

into n segments

then connecting the division points with all possible segments parallel

to the original sides, n² small triangles are created.

Proof by induction:

Let S = {nN: f(n) = n²}

1

Show 1 S:

f(n) =n²

f(1) = 1 = 1²

Show 2 S:

when dividing each side into 2 segments

and connecting division points as described,

4 small triangles are created.

f(n) =n²

f(2) = 4 = 2²

Show 3 S:

when dividing each side into 3 segments

and connecting division points as described,

9 small triangles are created.

f(n) =n²

f(3) = 9 = 3²

Assume n S.

Assume when dividing each side into n segments and connecting division points as described, n²

small triangles are created.

Assume f(n) = n².

Show n+1 S.

Show when dividing each side into n+1 segments and connecting

division points as described, (n+1)² small triangles are created.

Show f(n+1) = (n+1)².

Consider a divided triangle

with n segments on each side.

When a segment equal in size to the n segments is added to each side

and those endpoints are connected,

a space is created at the bottom of the original triangle.

Also, a new, bigger equilateral triangle has been created.

This new, bigger triangle has n+1 segments on each side.

n segments

+

1 segment

Now, the parallel dividing lines are extended down

to the base of the new, bigger triangle.

More small triangles are created.

The n segments of the base of the original triangle

correspond to n bases of the new, small triangles created.

Also, the n+1 segments of the base of the new, bigger triangle

correspond to n+1 bases of the new, small triangles.

So, n+(n+1) bases

correspond to n+(n+1) new, small triangles

By assumption, the original triangle has n segments on each side

And n² small triangles inside.

By adding 1 segment to each side of this triangle,

n + (n+1) small triangles are added.

The total small triangles of the new, bigger triangle is:

=n²+2n+1

=(n+1)(n+1)

n² + n +(n+1)

= (n+1)²

This shows n+1 S.

By induction, S N.

Dwight says, “that’s it.”