Finding the area of a circle tangentially inside a triangle

Solution 1:

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$AT = AP$

so $AT = 2$

Taking $B$ to be the origin of the Cartesian coordinate system, then

$A = (0, 3)$

The unit direction vector from A down the hypotenuse is $(\cos C, - \sin C) = (4/5, -3/5) $

$T = A + AT = A + 2 (4/5, -3/5) = (8/5, 9/5)$

$O = T + TO = (8/5, 9/5) + r (- 3/5, -4/5) = ( (8 - 3r)/5 , (9 - 4 r)/5 )$

$O$ is equidistance from $A$ and $B$ , hence $O$ lies on the perpendicular bisector of $AB$

thus,

$(9 - 4 r)/5 = 3/2$

$4 r = 3/2$

$r = 3/8$

Magenta Area = $\dfrac{ 9 \pi}{64} $