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Topic: Heart inscribed in a circle
Replies: 4   Last Post: Aug 25, 2010 5:04 AM

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Avni Pllana

Posts: 546
Registered: 12/6/04
Re: Heart inscribed in a circle
Posted: Aug 18, 2010 7:59 AM
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heart.png (11.3 K)

> 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.

Hi Stuart,

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),

y = x - b ... (4)

Finaly we have

angle(AOF) = atan(sqrt(2)/2)
~ 35.26439° .

Best regards,

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