Originally Posted by

**MarkFL2** I would use the Pythagorean theorem to state:

$\displaystyle (\bar{AP}+r)^2+(\bar{BP}+r)^2=(\bar{AP}+\bar{BP})^ 2$

$\displaystyle \bar{AP}^2+2r\bar{AP}+r^2+\bar{BP}^2+2r\bar{BP}+r^ 2=\bar{AP}^2+2\bar{AP}\bar{BP}+\bar{BP}^2$

$\displaystyle 2r\bar{AP}+2r\bar{BP}+2r^2=2\bar{AP}\bar{BP}$

$\displaystyle r\bar{AP}+r\bar{BP}+r^2=\bar{AP}\bar{BP}$

We are told $\displaystyle \bar{AP}\bar{BP}=24$ hence:

$\displaystyle r(\bar{AP}+\bar{BP}+r)=24$

Now, referring to your diagram, we may find the area $\displaystyle A$ of the triangle by adding together the square and 4 triangles making up the total triangle:

$\displaystyle A=r^2+r\bar{AP}+r\bar{BP}=r(\bar{AP}+\bar{BP}+r)$

Using our previous result, we find:

$\displaystyle A=24$