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 = {n  N: 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.”