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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

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.

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

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

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: