# Collinear Points Proved by Midpoint Theorem

In ∆XYZ, the medians ZM and YN are produced to P and Q respectively such that ZM = MP and YN = NQ. Prove that the points P, X and Q are collinear, and X is the midpoint of PQ.

Solution:

Given: In ∆XYZ, the points M and N are the midpoints of XY and XZ respectively. ZM and YN are produced to P and Q respectively such that ZM = MP and YN = NQ.

To prove: (i) P, X and Q are collinear.

(ii) X is the midpoint of PQ.

Construction: Join AX, XQ and MN.

Proof:

 Statement Reason 1. In ∆XPZ, M and N are the midpoints of PZ and XZ respectively. 1. Given. 2. Therefore, MN ∥ XP and MN = $$\frac{1}{2}$$XP. 2. By the Midpoint Theorem. 3. In ∆XQY, M and N are the midpoints of XY and YQ respectively. 3. Given. 4. Therefore, MN ∥ XQ and MN = $$\frac{1}{2}$$XQ. 4. By the Midpoint Theorem. 5. Therefore, XP ∥ MN and XQ ∥ MN. 5. From statements 2 and 4. 6. Therefore, XP and XQ lie in the same straight line. 6. Both passes through the same point X and are parallel to the same straight line MN. 7. Therefore, P, X and Q are collinear. [(i) Proved] 7. From statement 6. 8. Also, $$\frac{1}{2}$$XP = $$\frac{1}{2}$$XQ. 8. From statements 2 and 4. 9. Therefore, XP = XQ. 9. From statement 8. 10. Therefore, X is the midpoint of PQ. [(ii) Proved] 10. From statement 9.