Triangle Proof

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Triangle Proof

• by Kathy McDonald
• section 3.1 7

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Prove When dividing each side of an equilateral
triangle
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into n segments
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then connecting the division points with all
possible segments parallel
to the original sides, n² small triangles are
created.
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Proof by induction
Let S n?N f(n) n²
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Show 1 ?S
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f(n) n² f(1) 1 1²
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Show 2 ?S
when dividing each side into 2 segments
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and connecting division points as described,
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4 small triangles are created.
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f(n) n² f(2) 4 2²
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Show 3 ?S
when dividing each side into 3 segments
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and connecting division points as described,
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9 small triangles are created.
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f(n) n² f(3) 9 3²
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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.
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Show 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.
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Consider a divided triangle
with n segments on each side.
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When a segment equal in size to the n segments is
added to each side
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and those endpoints are connected,
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a space is created at the bottom of the original
triangle.
Also, a new, bigger equilateral triangle has
been created.
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This new, bigger triangle has n1 segments on
each side.
n segments

1 segment
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Now, the parallel dividing lines are extended
down
to the base of the new, bigger triangle.
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More small triangles are created.
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The n segments of the base of the original
triangle
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correspond to n bases of the new, small triangles
created.
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Also, the n1 segments of the base of the new,
bigger triangle
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correspond to n1 bases of the new, small
triangles.
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So, n(n1) bases
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correspond to n(n1) new, small triangles
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By assumption, the original triangle has n
segments on each side
And n² small triangles inside.
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By adding 1 segment to each side of this triangle,
n (n1) small triangles are added.
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The total small triangles of the new, bigger
triangle is
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n² n (n1)
n²2n1 (n1)(n1)
(n1)²
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This shows n1 ?S.
By induction, S ? N.
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Dwight says, thats it.