# 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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