> Has anyone constructed a heart shape (really square > with two semi-circles on two adjacent sides) inside a > given circle? The largest heart shape would be > tangent to the larger circle at a point on each of > the semicircles and then a vertex of the square > (opposite the two semicircles). Then can you > analytically find the angles? > > Thanks for your help.
Let A=[1 0], B=[0 1], C=[-1 0], D=[0 -1] be the vertices of the square, see attachment. The red circle with diameter AB makes the right 'hump' of the heart. Let the unit circle with center O and radius OA be the inversion circle, that maps the right 'hump' into the segment AB. In order to find the circumcircle of the heart, that passes through D and is tangent to the right 'hump', it is sufficient to find a circle that passes through D and is tangent to segment AB, since inversion preserves tangency. Let E=[0 -b] be the center of the sought circle, then we have
x^2 + (y+b)^2 = (1-b)^2 ... (1)
and for the line AB
x + y = 1 ... (2)
From the requirement that the system of equations (1),(2) must have only one solution, we obtain the following equation
(1+b)^2 - 8*b = 0 ... (3)
Solving (3) we obtain b = 3 - 2*sqrt(2) . We find point F = [2-sqrt(2), sqrt(2)-1], solving the system of equations (2) and (4),