2007 Alabama ARML TST Problems/Problem 15

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Problem

Let $P$ be a point inside isosceles right triangle $ABC$ such that $\angle C = 90^{\circ}$ , $AP = 5$, $BP = 13$, and $CP = 6\sqrt{2}$. Find the area of $ABC$.

Solution

2007AlabamaARMLTST15.png

Let $D$, $E$, and $F$ be the reflections of $P$ over sides $BC$, $CA$, and $AB$, respectively. We then have that $[AEC]=[APC]$, $[FAB]=[PAB]$, and $[BCD]=[BCP]$. This shows that $[AECDBF]=2[ABC]$. I shall now proceed to find $[AECDBF]$. This is equal to

\[[AECDBF]=[AEF]+[BDF]+[DEF]\]

Note that $\angle CAE=\angle CAP$ and $\angle BAP=\angle BAF$, so $\angle EAF=2\angle CAB=90^{\circ}$. Similarly, $\angle FBD=90^{\circ}$ and $\angle DCE=180^{\circ}$. Now note that $AE=AF=AP=5$ and $BD=BF=BP=13$. Therefore $[AEF]=\frac{25}{2}$ and $[BDF]=\frac{169}{2}$. Also note that $EF=5\sqrt{2}$ and $DF=13\sqrt{2}$. We also know that $D$, $C$, and $E$ are collinear, so $DE=EC+CD=2CP=12\sqrt{2}$. This shows that $DEF$ is a 5-12-13 right triangle, so it has area $\frac{EF\cdot DE}{2}=60$, so

\[[ABC]=\frac{[AECDBF]}{2}=\frac{\frac{25}{2}+\frac{169}{2}+60}{2}=\boxed{\frac{157}{2}}\]

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 14
Followed by:
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