MathDB

Suppose we have a square

ABCD

and a point

S

in the interior of this square. Under homothety with centre

S

and ratio of magnification

k > 1

, this square becomes another square

A'B'C'D'

. Prove that the sum of the areas of the two quadrilaterals

A'ABB'

and

C'CDD'

are equal to the sum of the areas of the two quadrilaterals

B'BCC'

and

D'DAA'

.
[asy] unitsize(3 cm);
pair[] A, B, C, D; pair S;
A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3);
draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed);
dot("

A

", A[0], N); dot("

B

", B[0], SE); dot("

C

", C[0], SW); dot("

D

", D[0], SE); dot("

A'

", A[1], NW); dot("

B'

", B[1], SW); dot("

C'

", C[1], SE); dot("

D'

", D[1], NE); dot("

S

", S, dir(270)); [/asy]

Problem Statement