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Problem of the Week #2 - April 9th, 2012

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Jameson

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Jan 26, 2012
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Thank you to Ackbach for this problem and to those of you who participated in last week's POTW!

Given a triangle $ABC$ and a point $D$ inside $ABC$, prove that $\overline{AD}+\overline{DC}\le \overline{AB}+\overline{BC}$.
 
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Jameson

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Jan 26, 2012
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Congratulations to the following members for their correct solutions:

1) caffeinemachine

Solution:

Extend line $AD$ such that it intersects with line $BC$ at point $E$. Use the triangle inequality twice:

$$\overline{AD}+\overline{DC}\le \overline{AD}+\overline{DE}+\overline{EC}$$

$$=\overline{AE}+\overline{EC}\le \overline{AB}+\overline{BE}+\overline{EC}$$

$$=\overline{AB}+\overline{BC}.$$

QED
 
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