Question

Prove that the three line segments which join the mid-points of the sides of triangle, divide it into four triangles which are congruent to each other.

Answer 1

Consider triangle ABC where M, N, and P are midpoints of sides AB, AC, and BC respectively.

From the Midpoint Theorem:

• MN || BC and MN = `1/2` BC
• NP || AB and NP = `1/2` AB
• MP || AC and MP = `1/2` AC

These three line segments MN, NP, and MP form triangle MNP inside triangle ABC, and also divide the triangle into four smaller triangles: AMN, BNP, CMP, and MNP.

We need to prove that these four triangles are congruent.

Proof:

Since M, N, P are midpoints:

• AM = MB, AN = NC, BP = PC (equal halves)

By construction, MN || BC and MN = `1/2` BC, NP || AB and NP = `1/2` AB, MP || AC and MP = `1/2` AC

Consider triangles AMN and CPN.

• AN = NC (since N is midpoint)
• AM = CP (since M and P are midpoints on sides)
• Angle MAN = Angle PCN (corresponding angles as MN || BC)

Hence, by SAS, △AMN ≅ △CPN.

Similarly, prove △BMP ≅ △CNP and △MNP ≅ the other two smaller triangles by corresponding sides and angles using parallel lines and midpoint equalities.

Therefore, the three segments joining the midpoints divide triangle ABC into four congruent triangles of equal area.