# Transversals' Theorem

## Famous theorems in 2D Geometry

**Theorem. **Whether triangle ABC and point D ∈ (BC), M ∈ (AB), N ∈ (AC) and {P} = MN ∩ AD. Then it has the relationship

**Proof.** We consider this figure:

We have two cases:

**First. (which is simple to proof)**

**Second. **

We build *d *|| BC, A ∈ *d* and we consider {X} = MN ∩ BC and {Y} = *d* ∩ MN

Becouse *d* || BC we have, according **Fundamental Theorem of Similarity**,

And then we get:

### Obs

If **P** is **centre of gravity** of the triangle we have

- D - mid of BC => BD = DC = BC / 2
- PD = 2 AD / 3 => PA = 2 • PD, i.e.:

So though, the relation

becomes

That means **P - **the centre of gravity of the triangle ABC, Transversals' Theorem becomes

**Keywords: **
theorem, transversal, 2D geometry, proof

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