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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} Show 1 S: 1 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 + n +(n+1) =n²+2n+1 =(n+1)(n+1) = (n+1)² This shows n+1 S. By induction, S N. Dwight says, “that’s it.”