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 4 of 5: Convert back to the requested ratios

r_i=\frac{PQ_i}{P_iQ_i}=\frac{PQ_i}{PP_i+PQ_i}\quad\Longrightarrow\quad\frac{PP_i}{PQ_i}=\frac{1-r_i}{r_i}

Detailed analysis

Since $P$ lies between $P_i$ and $Q_i$ , $P_iQ_i=PP_i+PQ_i$ . Solving the definition of $r_i$ gives $PP_i/PQ_i=(1-r_i)/r_i$ .