How can we prove that A', B', and C' are collinear in triangle ABC?

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The discussion centers on proving the collinearity of points A', B', and C' in triangle ABC, where A' is defined as a point on segment BC such that A'B is perpendicular to line PA. The participants clarify that the choice of A' does not influence the parallel condition with respect to PA, as points P, A, B, and C are fixed. The proof hinges on the geometric properties of perpendicularity and collinearity in triangle geometry.

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1 ) Let P be an arbitary point in the plane of triangle \vartriangle ABC , let A' be a point on BC such that A'B \bot PA
, Define B' , C' in the same way,P rove that A' , B' ,C' are collinear.
 
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I'm not sure the statement of the problem is correct; if A' is a point on segment BC, then A'B is a subsegment of BC, and the choice of A' does not affect at all the (possible or not) parallel condition with respect to PA, since all P, A, B and C are fixed beforehand.
 

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