# Proofs on Basic Proportionality

In this section, we shall discuss some more proofs on basic proportionality.**Theorem 1: The line drawn from the mid point of one side of a triangle parallel to another side bisects the third side.**

**Given :**Given ΔABC, D is the mid point of side AB. DE || BC.

**Prove that :**AE = EC.

statements |
Reasons |

1) DE || BC | 1) Given |

2) AD/DB = AE/EC | 2) By Basic Proportionality Theorem |

3) D is the mid point. | 3) Given |

4) AD = DB | 4) By definition of mid point. |

5) AD/DB = 1 | 5) By division property |

6) AE /EC = 1 | 6) From (2) and (5) |

7) AE = EC | 7) By cross multiply |

**Theorem 2 : Prove that the diagonals of a trapezoid(trapezium) divide proportionally.**

**Given :**ABCD is a trapezoid.

**Prove that :**DE/EB = CE /EA

**Construction :**Draw EF || BA || CD, meeting AD in F.

Statements |
Reasons |

1) FE || AB | 1) Given (in ΔABD) |

2) DE/EB = DF/FA | 2) By Basic proportionality theorem |

3) FE || DC | 3) Given (in ΔCDA) |

4) CE/EA = DF/FA | 4)By Basic proportionality theorem |

5) DE/EB = CE/EA | 5) From (2) and (4) |

**Theorem 3 : Any line parallel to the sides of a trapezium (trapezoid) divides the non-parallel sides proportionally.**

**Given :**ABCD is a trapezoid. DC || AB. EF || AB and EF || DC.

**Prove that :**AE/ED = BF/FC

**Construction :**Join AC, meeting EF in G.

Statements |
Reasons |

1) EG || DC | 1) Given (in ΔADC) |

2) AE/ED = AG/GC | 2) By Basic proportionality theorem |

3) GF || AB | 3) Given (in ΔABC) |

4) AG/GC = BF/FC | 4)By Basic proportionality theorem |

5) AE/ED = BF/FC | 5) From (2) and (4) |

**Similarity in Triangles**

• Similarity in Geometry

• Properties of similar triangles

• Basic Proportionality Theorem(Thales theorem)

• Converse of Basic Proportionality Theorem

• Interior Angle Bisector Theorem

• Exterior Angle Bisector Theorem

• Proofs on Basic Proportionality

• Criteria of Similarity of Triangles

• Geometric Mean of Similar Triangles

• Areas of Two Similar Triangles

• Similarity in Geometry

• Properties of similar triangles

• Basic Proportionality Theorem(Thales theorem)

• Converse of Basic Proportionality Theorem

• Interior Angle Bisector Theorem

• Exterior Angle Bisector Theorem

• Proofs on Basic Proportionality

• Criteria of Similarity of Triangles

• Geometric Mean of Similar Triangles

• Areas of Two Similar Triangles