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 5 of 5: Conclude the two inequalities

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r_i\le\frac13\Longrightarrow\frac{PP_i}{PQ_i}\ge2,\qquad r_j\ge\frac13\Longrightarrow\frac{PP_j}{PQ_j}\le2

Detailed analysis

The function $(1-r)/r$ decreases for positive $r$ . Thus $r_i\le1/3$ yields $PP_i/PQ_i\ge2$ , while $r_j\ge1/3$ yields $PP_j/PQ_j\le2$ . These are exactly the required one-not-smaller and one-not-larger ratios.