Geometry lesson in plan

Steiner's Theorem

Let's have the triangle ABC and the points M, N ∈(BC). If

Stein's theorem: MAB=NAC

then this relation is true:

Steiner's theorem, geometry, reports

Proof.

We use small geometric constructions. We build BE || AC, E ∈AM and CF || AB,  F  ∈ AN.

Steiner's Theorem, The figure, planic geometry

angles parallel sides (angles with parallel sides) (1)

(we know this from the hypothesys) (2)

From  (1) and (2) we have,

 

BE || AC => (acc. Fundamental Theorem of Similarity)   

CF || AB => (acc. Fundamental Theorem of Similarity)  

Steiner theorem, geometry

From the last two relations we get:

steiner theorem, geometry

 

 

 

Keywords: plan geometry theorems, theorem Steiner

 

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