Title: Triangle Proof
1Triangle Proof
- by Kathy McDonald
- section 3.1 7
2 Prove When dividing each side of an equilateral
triangle
3into n segments
4then connecting the division points with all
possible segments parallel
to the original sides, n² small triangles are
created.
5Proof by induction
Let S n?N f(n) n²
6Show 1 ?S
1
f(n) n² f(1) 1 1²
7Show 2 ?S
when dividing each side into 2 segments
8and connecting division points as described,
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114 small triangles are created.
12f(n) n² f(2) 4 2²
13Show 3 ?S
when dividing each side into 3 segments
14and connecting division points as described,
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179 small triangles are created.
18f(n) n² f(3) 9 3²
19Assume 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.
20Show n1 ?S.
Show when dividing each side into n1 segments
and connecting division points as described,
(n1)² small triangles are created. Show f(n1)
(n1)².
21 Consider a divided triangle
with n segments on each side.
22When a segment equal in size to the n segments is
added to each side
23and those endpoints are connected,
24a space is created at the bottom of the original
triangle.
Also, a new, bigger equilateral triangle has
been created.
25This new, bigger triangle has n1 segments on
each side.
n segments
1 segment
26Now, the parallel dividing lines are extended
down
to the base of the new, bigger triangle.
27More small triangles are created.
28The n segments of the base of the original
triangle
29correspond to n bases of the new, small triangles
created.
30Also, the n1 segments of the base of the new,
bigger triangle
31correspond to n1 bases of the new, small
triangles.
32So, n(n1) bases
33correspond to n(n1) new, small triangles
34By assumption, the original triangle has n
segments on each side
And n² small triangles inside.
35By adding 1 segment to each side of this triangle,
n (n1) small triangles are added.
36The total small triangles of the new, bigger
triangle is
37n² n (n1)
n²2n1 (n1)(n1)
(n1)²
38This shows n1 ?S.
By induction, S ? N.
39Dwight says, thats it.