As shown in the figure, circle O is the inscribed circle of triangle ABC, and the tangent points are D, e, F. the known angles BCA = 90 degrees, ad = 5cm, DB = 3cm. Find the area of triangle ABC D is on AB, e is on BC, f is on AC

Ad = AF = 5cm, BD = be = 3cm, CF = CE = radius (R)
(3 + R) ^ 2 + (5 + R) ^ 2 = 64 (according to Pythagorean theorem)
2×r^2+16r+34=64
2×r^2+16r-30=0
r=(-16±√(16^2+4×2×30))÷4
r=√31-4
Area = 1 / 2 × (3 + √ 31-4) × (5 + √ 31-4) = 15

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