# Can every triangle be divided into five isosceles triangles?

That's my problem: Can every triangle be divided into five isosceles triangles?

I've got to give evidence why this is true or not true...

(sorry for possible language mistakes - I'm from Germany)

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To summarize:

Hypothesis: Every triangle is "n-dissectable" (can be divided into n isosceles triangles) if $n \ge 4$.

Every triangle can be dissected into two right triangles (how?): by dissecting it along the altitude to its longest side. Each of these can be dissected into two isosceles triangles (why?): by cutting along the median from the right angle to the hypotenuse in each, giving a dissection of the original triangle into 4 isosceles triangles.

Now use induction on $n$ to prove the case that any triangle can be divided into 5 isosceles triangles.

To prove it draw a line from any of the vertex to opposite side to get triangles so that one of the triangles is isosceles. Consider the remaining triangle as the starting triangle and based on what we know about being 4-dissectable, that any triangle can be divided to 4 isosceles triangles, we get 4 isosceles triangles from this triangle, (the second of the two from the of original triangle. This gives us that the first isosceles triangle, plus the 4 sub-isosceles triangles combine to make any triangle dissectable (divided) into a total of 5 isosceles triangles.

An equilateral triangle can be dealt with separately: Cut off a small equilateral triangle from a corner (base parallel to to larger base), and since an isosceles trapezoid always has a circumcircle, one can obtain 4 isosceles triangles from within the trapezoind = total 5. [Alternatively, if you count the original equilateral triangle as a separate isosceles triangle, the proceed as in the case for $n = 4$ to obtain $4$ interior isosceles triangles, you have, in total, 5 isosceles triangles.]

The attachment can clarify matters further.

The above link goes into great detail and proves what you need to prove, and provides pictures that portray exactly what's going on, which might help, in terms of any language-translation barriers.

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What do you mean by induction on n? – user55214 Jan 4 '13 at 19:24
@user55214 You don't - you dissect your original triangle into an isosceles triangle $T_1$ and another (non-isosceles) triangle $T_2$, and then use the procedure outlined here to dissect $T_2$ into four isosceles triangles $T_{2a}, T_{2b}, T_{2c}, T_{2d}$. This then gives you a dissection of your original triangle into five isosceles triangles: $T_1, T_{2a}, T_{2b}, T_{2c},$ and $T_{2d}$. – Steven Stadnicki Jan 4 '13 at 19:35
Actually, one question now that I look at this: what if the original triangle is equilateral, so that all isosceles triangles that you can generate from it in the subdivision step leave trivial 'remaining triangles'? – Steven Stadnicki Jan 4 '13 at 19:40
@StevenStadnicki The equilaterial triangle has to be dealt with separately. It's simple to do so, Cut off a small equilateral triangle form the tip, and use the fact that an isosceles trapezoid always has a circumcircle to get 4 isosceles triangles. – Calvin Lin Jan 4 '13 at 20:05
Can isosceles be declared the same as equilateral? – user55214 Jan 4 '13 at 20:06

It is possible to divide a triangle into five isosceles triangles. This is provable by induction. The fact that all right triangles are 2-dissectible is important because every triangle can be divided into two right triangles by cutting along the altitude to its longest side. Each can then be cut into two isosceles triangles by cutting along the median from the right angle to the hypotenuse.

Hence using this we can prove that any triangle can be divided to 4 isosceles triangles. Further we use this to prove that any triangle can be divided to 5 isosceles triangles.

To prove it draw a line from any of the vertex to opposite side to get triangles. One of them will be isosceles. Consider other as new starting triangle and based on above proved theorem that any triangle can be divided to 4 isosceles triangles we get 4 isosceles triangles from other part of original triangle

Hence giving in total 5 isoceles triangles. Please feel free to ask if you are little confused over this description.

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Thank you for this great answer!!! I can mainly understand it, but as it's not German it's a little confusing ;) I will try to redo it on a sheet of paper and if there are any questions left I will just ask you :) – user55214 Jan 4 '13 at 19:36
In the first paragraph, I think you mean "by cutting along the median to its longest side". I do not understand what "To prove it draw a line from any of the vertex to opposite side to get triangles. One of them will be isosceles." means, but I can guess. You should elaborate more on it. – Calvin Lin Jan 4 '13 at 19:54