Gardner showed how to dissect a square into eight and nine acute scalene triangles.
W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, illustrated above (pers. comm. to Ed Pegg, Jr., Oct 25, 2002). The coordinates can be found from solving the four simultaneous equations
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(1)
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(2)
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(3)
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(4)
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for the four unknowns 
and picking the solutions for which 
. The solutions are roots of 12th order
polynomials with numerical values given approximately by
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(5)
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(6)
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(7)
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(8)
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Pegg has constructed a dissection of a square into 22 acute isosceles triangles.
Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).