Given:
∆ABC is an equilateral triangle, hence, all the three sides have the same length.
L, M, N are the midpoints of AC, CB, and AB. Hence, for instance, the distance between segment CM and MB are equal, by definition of midpoint.
Prove: LMNB is a rhombus.
Statement → Proof
1. ∆ABC is an equilateral triangle. → Given
2. Segment AB ≅ Segment AC ≅ Segment BC → Definition of an Equilateral Triangle
3. 1/2AB ≅ 1/2AC ≅ 1/2BC → Division Property of Equality
4. M and L are midpoints of BC and AC respectively. → Given
5. 1/2AB = Segment ML. → Midpoint Theorem
6. 1/2BC = Segment MB → Definition of Midpoint
7. Segment ML = Segment MB → Transitive Property of Equality using Statement 5 and 6
8. L and N are midpoints of AC and AB respectively. → Given
9. 1/2BC = Segment LN → Midpoint Theorem
10. 1/2AB = Segment BN → Definition of Midpoint
11. Segment LN = Segment BN → Transitive Property of Equality using Statement 9 and 10
12. Segment ML = Segment BN → Transitive Property of Equality using Statement 5 and 10
11. Segment MB = Segment LN → Transitive Property of Equality using Statement 6 and 9
13. Segment LN = Segment BN = Segment ML = Segment MB → Substitution Property of Equality using Statement 11 and 12
14. LMNB is a rhombus. → Definition of a rhombus.
One of the properties of a rhombus is that all 4 sides are equal in length.