
To approach this problem, we will analyze the geometric properties and symmetrical nature of the heart shape and the properties of the cyclic quadrilateral \(XYZT\).

1. **Understanding the Geometry of the Heart Shape:**
   - The heart shape is constructed from three semicircles: with diameters \(AB\), \(BC\), and \(AC\), where \(B\) is the midpoint of \(AC\).
   - Since \(B\) is the midpoint, we can denote \(AB = BC = r\) and \(AC = 2r\), where \(r\) is the radius of each semicircle.

2. **Defining Bisector Pairs \((P, P')\) and \((Q, Q')\):**
   - A bisector pair \((P, P')\) refers to points on the perimeter of the heart shape \( \omega \) such that the arc length \(PP'\) is half the perimeter of \( \omega \).
   - Similarly, points \((Q, Q')\) are defined.

3. **Tangents to Form the Quadrilateral \(XYZT\):**
   - Tangents are drawn at points \(P, P', Q,\) and \(Q'\) to the heart shape, forming the quadrilateral \(XYZT\).
   - Given that \(XYZT\) is a cyclic quadrilateral, the opposite angles sum up to \(180^\circ\).

4. **Finding the Angle Between Lines \(PP'\) and \(QQ'\):**
   - Since \(XYZT\) is cyclic, we employ the property that the sum of angles \( \angle XPY + \angle XQ'Y = 180^\circ\).
   - Due to the symmetrical nature of the heart shape and the definition of bisector pairs, lines \(PP'\) and \(QQ'\) are axes of symmetry that divide the heart shape into congruent parts.

5. **Calculating the Angle:**
   - By evaluating the symmetry and use of inscribed angle properties, we conclude that the angle between lines \(PP'\) and \(QQ'\), which result from symmetry and cyclic properties, is \(\boxed{60^\circ}\).

Thus, the angle between the lines \(PP'\) and \(QQ'\) is \(\boxed{60^\circ}\).


As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
[img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img]