State and Prove Mid-point theorem.

The Midpoint Theorem is a fundamental theorem in geometry, particularly in the study of triangles. Here's the statement and proof of the theorem:

Statement of Midpoint Theorem: In a triangle, the line segment joining the midpoints of any two sides will be parallel to the third side and will be half its length.

Proof of Midpoint Theorem:

Let's consider a triangle ABC. Let D and E be the midpoints of AB and AC respectively. We need to prove that DE is parallel to BC and DE = 1/2 BC.

Step 1: Draw the line segment DE.

Step 2: By the properties of a triangle, we know that triangle ADE is similar to triangle ABC (AA rule). Therefore, the sides are proportional, i.e., AD/AB = DE/BC.

Step 3: But D is the midpoint of AB, so AD = 1/2 AB.

Step 4: Substituting this into the proportion gives us 1/2 = DE/BC.

Step 5: Therefore, DE = 1/2 BC.

Step 6: Also, by the properties of similar triangles, the corresponding angles are equal. So, angle ADE = angle ABC and angle AED = angle ACB.

Step 7: But if two lines are cut by a transversal such that the alternate angles are equal, then the lines are parallel (Alternate Angle Axiom).

Step 8: Therefore, DE is parallel to BC.

Hence, the Midpoint Theorem is proved.