We can place two pairs of similar triangles within the shape as shown above. The red triangles are clearly in the ratio of 3,4,5. It’s less obvious what the ratio of the sides of the blue triangles is, until you construct a grid as below which demonstrates that if you want to add an angle to the small angle of a 3,4 right triangle to total 45 degrees, that new triangle will have legs in the ratio of 7 to 1. (This generalises to a triangle with legs a,b pairing with a triangle with legs a+b,|a-b|).

So c/b=7, c/a=4/3 and a+b=1. From this we can work out that a=21/25, b=4/25 and c=28/25. By Pythagoras, d is therefore 4/5*sqrt(2). By similar triangles, e/5=c/d, so e=7/2*sqrt(2).

Since the triangle is right-angled and isosceles, its area is ((d+e)^2)/2, which equals exactly 18.49.