Solution of the Week #442 - Triangle on a Square

A trial and error method is a good way to approach this particular puzzle, plugging in different potential values for DE, calculating DF from the similar triangles DEF and CEB, then calculating the area and perimeter of DEF. You will soon find that DE=8, DF=6, and the area and perimeter of DEF is 24.

For a more systematic approach, I used the nice fact, easily proved, that a triangle has the same area and perimeter if and only if the radius of its incircle is equal to 2. So putting in place a coordinate system where B is the origin, I place a circle of radius 2 at (22,26). Then, because I need EFB will be tangent to this circle, I construct another circle, centred on (11,13) and passing through the origin. Using the formulae of the two triangles to work out where they coincide*, I can find the point on line EFB has position (20.4,27.2). This enables me to find the position of point E to be (24,32), and the area and perimeter of triangle DEF to be 24.