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Isosceles from Regular (Posted on 2012-01-20) Difficulty: 3 of 5
Which regular polygons can be dissected into isosceles triangles by non-intersecting diagonals?

  Submitted by Jer    
Rating: 5.0000 (1 votes)
Solution: (Hide)
First off any even sided regular polygon can be converted into one of half as many sides by connecting every other vertex.

This mean that since a square can be done, every number of sides of the firm 2n can also be done.

It also means we now only have to consider the odd numbers, since the rest of the evens can be converted to an odd by this process.

An equilateral triangle is isosceles so we get this one for free as well as all numbers of the form 3*2n

The next cool thing is that if you connect any two diagonals that are a power of 2 away from each other you can convert the rest of this side into isosceles triangles by the same process.

There is only one type of isosceles triangle left to mention: the one on odd sided polygons that connects two consecutive vertices to the one directly opposite.

This leaves two equal sides, which if they are powers of two will be reducible as well. In other words numbers of the form 2m + 1 are doable as well as any multiple of this by a power of 2.

So the final answer is any number of the form (2m + 1)*2n for any non-negative integers n,m.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): Equilateral triangle are certainly isosceles.Dustin2012-01-22 01:34:29
re(2): Equilateral triangle are certainly isosceles.Jer2012-01-21 23:11:16
re: Equilateral triangle are certainly isosceles.Dustin2012-01-21 16:55:03
Equilateral triangle are certainly isosceles.Jer2012-01-21 16:18:13
re: Ideas / Partial solution?Larry2012-01-21 15:59:43
Does equilateral count as isosceles?Larry2012-01-21 10:08:44
Ideas / Partial solution?Dustin2012-01-21 05:48:20
Some Thoughtsa conjectureDej Mar2012-01-21 04:20:41
re: Another n-gonbrianjn2012-01-21 02:49:14
Another n-gonDustin2012-01-21 02:43:21
re(2): Not sure if this is what you want - A rewrite.brianjn2012-01-21 02:16:56
re: Not sure if this is what you want - A rewrite.Jer2012-01-21 00:05:43
Not sure if this is what you wantLarry2012-01-20 23:18:38
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