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 1 of 5: Use the centroid and the median ratios

Previous step Next step

G= A_1B_2\cap A_2B_3\cap A_3B_1,\qquad A_3G:GB_1=2:1

Detailed analysis

The points $B_n$ are the side midpoints, so $A_nB_{n+1}$ are the medians and meet at the centroid $G$ . On each median the centroid divides the segment in the ratio $2:1$ ; for example, if $A_3B_1=21t$ , then $A_3G=14t$ and $GB_1=7t$ .