Problem 4

Inside triangle

P_1P_2P_3

a point

P

is given. Let

Q_1,Q_2,Q_3

be the intersections of

PP_1,PP_2,PP_3

with the opposite sides. Prove that among

\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}

there is one not larger than

2

and one not smaller than

2

Step 2 of 5: Sum the three area ratios

r_1+r_2+r_3=\frac{[PP_2P_3]+[PP_3P_1]+[PP_1P_2]}{[P_1P_2P_3]}=1

Detailed analysis

The three subtriangles partition $P_1P_2P_3$ , so their areas add to the area of the whole triangle. Therefore $r_1+r_2+r_3=1$ .