# Midpoint Theorem by using the Equal Intercepts Theorem

Here we will prove that converse of the Midpoint Theorem by using the Equal Intercepts Theorem.

Solution:

Given: P is the midpoint of XY in ∆XYZ. PQ ∥ YZ.

To prove: XQ = QZ.

Construction: Through X, draw MN ∥ YZ.

Proof:

 Statement Reason 1. PQ ∥ YZ. 1. MN ∥ YZ and PQ ∥ YZ. 2. MN ∥ PQ ∥ YZ. 2. XP = PY. 3. The transversal XZ also makes equal intercepts XQ and QZ on MN, PQ and YZ. 3. By the Equal Intercepts Theorem. 4. Therefore, XQ = QZ. (Proved) 4. From statement 3.

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