Circumscriptible_Construction

Construction of circumscriptible quadrangles

To construct quadrangles with given side-lengths a, b, c, d such that a+c = b+d = t.

Construct first a segment t = (AB) and put points C, D (pressing, while defining the Shift key) such that a = (AC), c = (CB), b = (AD), d = (DB). Define three new points E, I, J (to be used to control the shape of the quadrangle). Transfer the lengths as shown. Modifiable are A, B, E, I, H (after pressing Ctrl+1) and C, D later two after pressing (Ctrl+2).

All these quadrangles are circumscriptible. The incircle is defined by the equation
r^2 = (a*u*z+c*x*y)/(a+c), where x, y, z, u are determined by solving the equations:
x+y = a,
y+z = b,
z+u = c,
u+x = d.
x can be taken (almost) free. Then y = a-x, z = b-a+x, u = d-x. The compatibility condition for the linear system is just: a+c = b+d. Replacing y, z, u in the previous formula gives a function of x:
r^2 = (a*d*(2*x+b-a))/(a+c) - x^2 .
Look at Circumscriptible.html for a geometric proof of the sufficiency of the condition a+c = b+d.

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