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Philippine Mathematical Olympiad

Geometry Difficulty 5.0 AIME, harder Prove it Philippines

Problem:

A triangle $ABC$ is to be constructed so that $A$ is at $(3,2)$ , $B$ is on the line $y = x$ , and $C$ is on the $x$ -axis. Find the minimum possible perimeter of $\triangle ABC$ .

Solution

Solution:

Let $D(2,3)$ be the reflection of $A$ with respect to $y = x$ , and $E(3,-2)$ the reflection of $A$ with respect to the $x$ -axis. Then if $B$ is on $y = x$ and $C$ is on the $x$ -axis, then $|AB| = |DB|$ and $|AC| = |CE|$ . Then the perimeter of $\triangle ABC$ is
$|AB| + |BC| + |AC| = |DB| + |BC| + |CE| \geq |DE| = \sqrt{26}$
Clearly, this lower bound can be attained.

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