Problem 3

Let

A_1,A_2,A_3

be three points in the plane, with

A_4=A_1

and

A_5=A_2

. For

n=1,2,3

, let

B_n

be the midpoint of

A_nA_{n+1}

and

C_n

the midpoint of

A_nB_n

. Let

D_n=A_nC_{n+1}\cap B_nA_{n+2}

and

E_n=A_nB_{n+1}\cap C_nA_{n+2}

. Calculate the ratio of the area of

D_1D_2D_3

to the area of

E_1E_2E_3

Step 5 of 5: Take the ratio of the two homotheties

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\frac{[D_1D_2D_3]}{[E_1E_2E_3]}=\left(\frac{2/7}{2/5}\right)^2=\boxed{\frac{25}{49}}

Detailed analysis

The linear ratio from the $E$ triangle to the $D$ triangle is $(2/7)/(2/5)=5/7$ . Areas scale as the square of a linear ratio, giving $[D_1D_2D_3]/[E_1E_2E_3]=25/49$ .