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The proof below is an indirect proof based on the
work of Conway dating from around 1995.

PS file to the whole proof

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From the above figure we know that 3a+3b+3c = 180o, which means that a+b+c= 60o.

Let x+ = x+60o for any angle x. since a+b+c = 60o. Let 0+=a+b+c=60o., This shows that it is possible to construct an triangle with 3 different types of angle combinations:
Type 1: 0+,0+,0+;
Type 2: a,b+,c+; a+,b,c+; a+,b+,c
Type 3: a++,b,c; a,b++,c; a,b,c++

since these seven combinations of angles all have a sum of 180o.

Instead of working forward, Conway worked backwards. He showed that from an equilateral triangle one can construct a triangle with any angles, i.e. with arbitrary a,b and c (that sums to 60o). According to Conway, we can make the following constructions:

Type 1. Construct an equilateral triangle with length 1.

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Type 2. Construct a triangle with the side joining larger angles ( e.g. a+ and b+) to have length 1. for example:

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Type 3. Construct two lines that intersects the side opposite to b++ at angle b+, thus forming an isosceles triangle with base angle =b+.

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The significance of constructing the isosceles triangle is to prove JHI and DFC are congruent.

1.JHI = DFC = b+

2. JIH = DCF = c

3. By construction, we get JH = DF=1

The above proves that JHI andDFC are congruent. This result is important because it shows that DC = JI are equal, when the two triangles are matched together, they become the common edge, then point J = point D and point I = point C.

Other triangles are constructed in similar way, and we can get the following:

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Note that the angles at the vertex of the equilateral triangle (e.g. D) sums up to 360o.

Check: (a+)+(c+)+(b++)+(0+)
= a+60o+c+60o+b+120o+60o

Since the angles at D sums up to 360o, and since the sides match due to congruency, we see that the triangles can be assembled perfectly together to form a larger triangle with angles 3a, 3b and 3c, and hence the conclusion.

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