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 2 of 5: Apply Menelaus to locate D1

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D_1=A_1C_2\cap B_1A_3\Rightarrow D_1B_1=3t,\ A_3D_1=18t,\ GD_1=4t

Detailed analysis

In triangle $B_1A_2A_3$ , the line $A_1D_1C_2$ meets its sides at the indicated midpoint-related points. Menelaus' theorem, with the midpoint ratios, gives $D_1B_1/A_3B_1=3/21$ . Hence $D_1B_1=3t$ and $A_3D_1=18t$ , so $GD_1=A_3D_1-A_3G=4t$ .