Dissecting Triangles into Isosceles Triangles - Canadian ...

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Suppose no cut goes fromavertex to the opposite side. The only possible
conguration is the one illustrated in Figure 1(a). Since the three angles
at this point sum to 360 , at least two of them must be obtuse. It follows
that the three arms have equal length and this point is the circumcentre of
the original triangle. Since itisaninterior point, the triangle is acute. Thus
all acute triangles are 3-dissectible.
In all other cases, one of the cuts gofromavertex to the opposite side,
dividing the triangle into an isosceles one and a 2-dissectible one. There are
quite a number of cases, but the argument is essentially an elaboration of
that used to determine all 2-dissectible triangles. We leave the details to the
reader, and will just summarize our ndings in the following statement.
Theorem.
A triangle is 3-dissectible if and only if it satises at least one of the following
conditions:
1. It is an isosceles triangle.
2. It is an acute triangle.
3. It is a right triangle.
4. It has a 45 angle.
5. It has one of the following forms:
(a) (, 90 , 2, 90 + ), 0