**Proof:**

To prove that S is the midpoint of YZ in triangle XYZ, we can use the properties of an isosceles triangle and the angle bisector theorem.

**1. Given Information:**
- Triangle XYZ is an isosceles triangle.
- XY = XZ.
- XS bisects angle YXZ and meets YZ at point S.

**2. Definition of an Isosceles Triangle:**
In an isosceles triangle, the two sides opposite the equal angles are congruent. In triangle XYZ, since XY = XZ, we can conclude that angle Y and angle Z are equal.

**3. Angle Bisector Theorem:**
According to the angle bisector theorem, if a line segment bisects an angle in a triangle, it divides the opposite side into two segments that are proportional to the adjacent sides. In triangle XYZ, XS bisects angle YXZ, so we can write:

YS / SZ = XY / XZ

**4. Substitution:**
Since XY = XZ (given), we can substitute the values in the equation:

YS / SZ = 1

**5. Conclusions:**
From the equation YS / SZ = 1, we can conclude that YS = SZ. This implies that point S is equidistant from both Y and Z, making it the midpoint of YZ.

**6. Proof by Contradiction:**
Alternatively, we can prove that S is the midpoint of YZ by contradiction. Let's assume that S is not the midpoint of YZ and that S divides YZ into two segments, SY and SZ, such that SY ≠ SZ. In this case, one of the segments would be longer than the other.

**7. Contradiction:**
Now, let's consider the angle bisector XS. Since XS bisects angle YXZ, it divides YZ into two segments, YS and SZ. However, since YS ≠ SZ (according to our assumption), XS cannot be the angle bisector.

**8. Contradiction Resolution:**
As a result of our contradiction, we conclude that our assumption was incorrect. Therefore, S must be the midpoint of YZ.

**9. Conclusion:**
Both the angle bisector theorem and the proof by contradiction demonstrate that S is the midpoint of YZ in triangle XYZ.