Squares in triangles
Source:
March 16, 2006
geometryincenterratio
Problem Statement
Squares and are inscribed in right triangle , as shown in the figures below. Find if area and area.[asy]
size(250);
real a=15, b=5;
real x=a*b/(a+b), y=a/((a^2+b^2)/(a*b)+1);
pair A=(0,b), B=(a,0), C=origin, X=(y,0), Y=(0, y*b/a), Z=foot(Y, A, B), W=foot(X, A, B);
draw(A--B--C--cycle);
draw(W--X--Y--Z);
draw(shift(-(a+b), 0)*(A--B--C--cycle^^(x,0)--(x,x)--(0,x)));
pair point=incenter(A,B,C);
label("", A, dir(point--A));
label("", B, dir(point--B));
label("", C, dir(point--C));
label("", (A.x-a-b,A.y), dir(point--A));
label("", (B.x-a-b,B.y), dir(point--B));
label("", (C.x-a-b,C.y), dir(point--C));
label("", (x/2-a-b, x/2));
label("", intersectionpoint(W--Y, X--Z));
dot(A^^B^^C^^(-a-b,0)^^(-b,0)^^(-a-b,b));[/asy]