118. An usual remarkable geometrical locus.

by Virgil Nicula, Sep 12, 2010, 11:46 PM

You can use the following remarkable geometrical locus ! It is very usually in the construct problems.

Lemma. Given are a triangle $ ABC$, a line $ d$ and a point $ F\not\in d$ . For a mobile point $ M\in d$ construct the triangle $ FML\sim ABC$

(in fact, are two triangles $ FML'\sim FML''\sim ABC$, where the points $ L'$, $ L''$ are symmetrically w.r.t. the line $ FM$). Define the

fixed points $ \{M_{1}, M_{2}, L_{1},L_{2}\}\subset d$ (in fact, the points $ M_{1}$, $ M_{2}$ are particular positions of the mobile point $ M$ and the points $ L_{1}$, $ L_{2}$

are particular positions of the points $ L'$, $ L''$ respectively) so that $ FM_{1}L_{1}\sim FM_{2}L_{2}\sim ABC$ ( i.e. $ \widehat{FL_{1}L_{2}}\equiv\widehat{FL_{2}L_{1}}\equiv\widehat{ACB}$)

and the fixed point $ R$ so that the line $ d$ separates the points $ P$, $ R$ and $ \widehat{L_{1}L_{2}R}\equiv\widehat{L_{2}L_{1}R}\equiv\widehat{BAC}$.

Then the geometrical locus $ \Lambda$ of the point $ L$ is a reunion of two lines, more exactely $ \Lambda = RL_{1}\cup RL_{2}$.

Hint.
This post has been edited 7 times. Last edited by Virgil Nicula, Nov 23, 2015, 7:44 AM

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