Find length of rectangle?

Rectangle is split, by diagonal, in two right triangles, and in bought are circles inscribed.

Distance of centers of the circles is 1, and width of rectangle is also 1.

Find length of rectangle a=?

? Favorite Answer

If r = in-radius of the circle, from the traingle,

r = area of traingle / semi-perimeter

=> r = a/[1 + a + √(1+a^2)] … (1)

Constructing a right triangle with a line joining the centers of the circles as hypotenuse and sides parallel to sides of the rectangle,

(1 – 2r)^2 + (a – 2r)^2 = 1 … (2)

Rationalizing the value of r in eqn. (1),

=> r = a[1 + a – √(1+a^2)] / [1 + a + √(1+a^2)] * [1 + a – √(1+a^2)]

=> r = (1/2) [1 + a – √(1+a^2)] => 2r = 1 + a – √(1+a^2)

=> 1 -2r = √(1+a^2) – a

and a – 2r = √(1+a^2) – 1

Plugging in eqn. (2),

[√(1+a^2) – a]^2 + [√(1+a^2) – 1]^2 = 1

=> 2(1 + a^2) + a^2 – 2(a + 1)√(1+a^2) = 0

=> (3a^2 + 2) = 2(a + 1)√(1+a^2)

Squarring,

9a^4 + 12a^2 + 4 = 4(a^2 + 2a + 1)(1 + a^2)

=> 9a^4 + 12a^2 + 4 = 4(a^4 + 2a^3 + 2a^2 + 2a + 1)

=> 5a^4 – 8a^3 + 4a^2 – 8a = 0

=> 5a^3 – 8a^2 + 4a – 8 = 0 … [because a ≠ 0]

Putting in Wolfram Alpha (link 1) gives

a ≈ 1.68771

and its exact value as

a = (8/15) + (1/15)*[2492 – 3√(69)]^(1/3)]

+ (1/15)[623 + 75√(69)]^(1/3)].

Putting the equations (1) and (2) in wolfram alpha (link 2)

a ≈ 1.68771 and r ≈ 0.362993

For exact values of a and r, click on exact forms in the link.

What I have is a = 1.687710482194933

I’m not having much luck in reducing the expression for this value, which is the one real root of the cubic equation:

5a³ – 8a² + 4a – 8 = 0