A heart shape is formed by constructing two unit semicircles on consecutive sides of a square of side length 2.
Find the radius of the circumcircle of this shape.
I don't have an exact solution, but here is a graphical answer with the help of Desmos:
https://www.desmos.com/calculator/wcpx1ju2nc
Let the square be positioned at (±1, ±1)
The half circles are:
(x1)^2 + y^2 = 1 with x>1 and
x^2 + (y1)^2 = 1 with y>1
The circumcircle is centered at (a,a)
and has radius √2*(1+a)
(xa)^2 + (ya)^2 = 2(1+a)^2 < Eqn of Circumcircle
whose radius is (√2)*(1+a)
Using Desmos to adjust a gives a ~ 0.262
and radius of circumcircle ~ 1.785
I have not yet found an exact algebraic solution for a or the radius, but I did get the equation for the tangent line in terms of a.
Take the equation for one of the half circles and the equation for the circumcircle.
x^2 + (y1)^2 = 1
(xa)^2 + (ya)^2 = 2(1+a)^2
x^2 + y^2  2y + 1 = 1
x^2 + y^2 = 2y
x^2  2ax + a^2 + y^2  2ay + a^2 = 2a^2 + 4a + 2
2y  2ax  2ay = 4a + 2
2y  2 = 4a + 2ax + 2ay
y  1 = 2a + ax + ay
y(1a) = ax + 2a + 1
y = (a/(1a))*x + (2a+1)/(1a)
This line is tangent to the intersection of the half circle and circumcircle.
The slope of the tangent line is a/(1a)

Posted by Larry
on 20240305 12:02:51 