geometry puzzle (12)

In the figure below we notice right-angled triangles and their inscribed circles.



Question: what is the area of the rectangle for these inscribed circle diameters?

Please look at the picture below:



Triangles DEC, CEB and BAD are similar.
See equal angles marked by the same symbols.
Affine lines in the triangles have the same ratio.
So we know that the hypothenuses have ratios 3:4:5,
the same as the diameters of the inscribed circles.

Now define:
    AB=3x
    BC=4x
    BD=5x
which is in accordance with the Pythagoras lemma.

The inscribed circle of triangle ABD has a radius of 2,5.
The area of triangle ABD is (base x height / 2)
    0,5 . 2,5 . 3x + 0,5 . 2,5 . 4x + 0,5 . 2,5 . 5x = 15x.
The area of triangle ABD also is:
    0,5 . 4x . 3x = 6x2
from
    15x = 6x2
follows
    x = 2,5
The area of rectangle ABCD is:
    3 . 2,5 . 4 . 2,5 = 75


Appendix




Writing the area of triangle ABC as [ABC]
we find for the radius r of the inscribed circle (see picture above):
    0,5ra + 0,5rb + 5,5rc = [ABC]
so:



Could this formula be simplified given that LA = 900?
In that case we know:
    a2 + b2 = c2
    2[ABC] = ab


Let's go: