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 3 of 5: The D triangle is a homothetic copy

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\frac{GD_1}{GA_3}=\frac{4t}{14t}=\frac27

Detailed analysis

The construction is cyclic in the indices, so the same computation gives $GD_n/GA_n=2/7$ for $n=1,2,3$ . Therefore $D_1D_2D_3$ is homothetic to $A_1A_2A_3$ with center $G$ and ratio $2/7$ .