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smc

geometry senior

Problem

Let

O

be an interior point of triangle

A B C

, and let

s_{1} = O A + O B + O C

. If

s_{2} = A B + B C + C A

, then

(A)

f or e v er y t r ian g l e s_{2} > 2 s_{1} s_{1} l t . e q s_{2}

(B)

f or e v er y t r ian g l e s_{2} > 2 s_{1} s_{1} < s_{2}

(C)

f or e v er y t r ian g l e s_{1} > 1/2 s_{2} s_{1} < s_{2}

(D)

f or e v er y t r ian g l e s_{2} g t . e q 2 s_{1} s_{1} l t . e q s_{2}

Solution

By the Triangle Inequality, we see that

s_{1} > \frac{1}{2} s_{2}

, therefore

for every triangle s_{1} > \frac{1}{2} s_{2}, s_{1} < s_{2}

.

Final answer

C