jmc

Let the triangle have coordinates $(0,0),(12,0),(0,5).$ Then the coordinates of the incenter and circumcenter are $(2,2)$ and $(6,2.5),$ respectively. If we let $M=(x,x),$ then $x$ satisfies$$\sqrt{(2.5-x{)}^2+(6-x{)}^2}+x=6.5$$$$2.5^2-5x+x^2+6^2-12x+x^2=6.5^2-13x+x^2$$$$x^2=(5+12-13)x$$$$x\ne 0⟹x=4.$$Now the area of our triangle can be calculated with the Shoelace Theorem. The answer turns out to be $\boxed{\frac{7}{2}}$