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 4 of 5: Apply Menelaus to locate the E triangle

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E_1=A_1B_2\cap C_1A_3\Rightarrow\frac{GE_1}{GA_1}=\frac25

Detailed analysis

Applying Menelaus to triangle $A_1A_2B_2$ and the line $C_1E_1A_3$ , using the midpoint ratios, gives $GE_1/GA_1=2/5$ . Cyclic symmetry gives $GE_n/GA_n=2/5$ for all three indices, so $E_1E_2E_3$ is homothetic to $A_1A_2A_3$ with ratio $2/5$ .