No, your friend is not correct. The statement "PQ = QR" does not necessarily imply that Q is the midpoint of segment PR.

Explanation:

• PQ = QR means that the distances between $P$ and $Q$, and $Q$ and $R$, are equal. This suggests that $Q$ divides the segment $PR$ into two equal parts, but this is only true if $P$, $Q$, and $R$ are collinear (i.e., they lie on the same straight line).

• If $P$, $Q$, and $R$ are not collinear, then $Q$ may not lie on the line segment $PR$, meaning $Q$ would not be a midpoint. In such a case, even though the distances $PQ$ and $QR$ are equal, $Q$ would not divide the segment $PR$ in half.

Thus, for $Q$ to be the midpoint of segment $PR$, the points must be collinear in addition to having equal distances. Therefore, $PQ = QR$ alone is not enough to conclude that $Q$ is always the midpoint of $PR$.

Do you have any questions, or would you like more details?

Here are 5 related questions for further exploration:

1. What is the definition of a midpoint in geometry?
2. How can we prove that a point is the midpoint of a line segment?
3. What other conditions must hold for $PQ = QR$ to imply that Q is the midpoint?
4. What happens if $PQ = QR$ but the points are not collinear?
5. How does the concept of vectors help to prove midpoint relationships?

Tip: When solving geometric problems, always check for additional conditions like collinearity or specific orientations of the points.