Shifting Circular Heart

Relevant wiki: Pythagorean Theorem

If each circular arc is tangential to its respective circle, then the centre of the arc must be on the opposite side of the circle from their tangential point, because the radius of the arc is equal to the diameter of the circle.

With this, we know that the maximum distance from the centre of a circle to any point on its respective arc is equal to the radius of a circle ( $\color{#D61F06}r$ ) plus the radius of an arc ( $\color{#20A900}R$ ).

$\color{#D61F06}r \color{#333333}+ \color{#20A900}R \color{#333333}= 3$

$3$ is then equal to the maximum length of a hypotenuse of a right triangle; formed from the centre of a circle, the tangent of the two circles, and the intersection of the two arcs.

Using the Pythagorean Theorem, we can determine the maximum length ( $\color{#3D99F6}m$ ) between the tangent of the two circles and the intersection of the two arcs.

$3^2 = 1^2 + \color{#3D99F6}m^2$

$\sqrt{3^2 - 1^2} = \color{#3D99F6}\sqrt{8} \color{#333333}= \boxed{\large\color{#3D99F6}2.828}$