Let $A, B, C, D$ be four distinct points in the plane such that the length of the six line segments $AB, AC, AD, BC, BD, CD$ form a $2$-element set ${a, b}$. If $a > b$, determine all the possible values of $\frac ab$.
1. **Applying the Pigeonhole Principle:**
   Given that the lengths of the six line segments \(AB, AC, AD, BC, BD, CD\) form a 2-element set \(\{a, b\}\) with \(a > b\), by the pigeonhole principle, at least three of these segments must have the same length. Let's assume without loss of generality that three of these segments have length \(a\).

2. **Case 1: Three segments of length \(a\) form a triangle:**
   - Suppose \(AB = AC = AD = a\). This implies that \(A\) is equidistant from \(B\), \(C\), and \(D\). Therefore, \(A\) forms an equilateral triangle with \(B\), \(C\), and \(D\).
   - The remaining segments \(BC\), \(BD\), and \(CD\) must all be of length \(b\). This configuration forms an equilateral triangle with side length \(b\).
   - The ratio \(\frac{a}{b}\) in this case is \(\sqrt{3}\) because the side length of an equilateral triangle is related to the height by a factor of \(\sqrt{3}\).

3. **Case 2: Three segments of length \(a\) form a square:**
   - Suppose \(AB = BC = CD = a\). This implies that \(A\), \(B\), \(C\), and \(D\) form a square.
   - The diagonals \(AC\) and \(BD\) will have length \(b\), where \(b = a\sqrt{2}\).
   - The ratio \(\frac{a}{b}\) in this case is \(\frac{a}{a\sqrt{2}} = \frac{1}{\sqrt{2}} = \sqrt{2}\).

4. **Case 3: Three segments of length \(a\) form a pentagon:**
   - Suppose \(AB = BC = CD = a\) and the remaining segments \(AD\) and \(BD\) are of length \(b\).
   - This configuration forms a regular pentagon where the ratio \(\frac{a}{b}\) is \(\frac{1 + \sqrt{5}}{2}\), which is the golden ratio.

5. **Case 4: Three segments of length \(a\) form a different configuration:**
   - Suppose \(AB = AC = AD = a\) and the remaining segments \(BC\), \(BD\), and \(CD\) are of length \(b\).
   - This configuration forms a tetrahedron where the ratio \(\frac{a}{b}\) is \(\sqrt{3}\).

6. **Case 5: Other configurations:**
   - If \(AB = AC = AD = a\) and \(BC = BD = CD = b\), then the ratio \(\frac{a}{b}\) is \(\sqrt{3}\).
   - If \(AB = BC = CA = a\) and the remaining segments \(AD\), \(BD\), and \(CD\) are of length \(b\), then the ratio \(\frac{a}{b}\) is \(\frac{1}{\sqrt{2 - \sqrt{3}}}\).

Conclusion:
The possible values of \(\frac{a}{b}\) are \(\sqrt{2}\), \(\sqrt{3}\), \(\frac{1 + \sqrt{5}}{2}\), and \(\frac{1}{\sqrt{2 - \sqrt{3}}}\).

The final answer is \( \boxed{ \sqrt{2}, \sqrt{3}, \frac{1 + \sqrt{5}}{2}, \frac{1}{\sqrt{2 - \sqrt{3}}} } \).