Task 201

The task

Triangle $ABC$ and triangle $DEF$ are congruent, isosceles right triangles. The square inscribed in triangle $ABC$ has an area of 15 square centimeters. What is the area of the square inscribed in triangle $DEF$? Express your answer as a common fraction. [asy] draw((0,0)--(2,0)--(0,2)--(0,0)--cycle,linewidth(2)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)--cycle,linewidth(2)); draw((1/8,0)--(1/8,1/8)--(0,1/8),linewidth(1)); draw((3,0)--(5,0)--(3,2)--(3,0)--cycle,linewidth(2)); draw((3+2/3,0)--(3+4/3,2/3)--(3+2/3,4/3)--(3,2/3)--(3+2/3,0)--cycle,linewidth(2)); draw((3+1/8,0)--(3+1/8,1/8)--(3,1/8),linewidth(1)); label("A",(0,2),NW); label("B",(0,0),SW); label("C",(2,0),SE); label("D",(3,2),NW); label("E",(3,0),SW); label("F",(5,0),SE); [/asy]