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line_parallel_to_line_through_point

line_parallel_to_line_through_point(l,m,P) states that l is the line parallel to line m that passes through point P.

Formal properties

  • line_parallel_to_line_through_point(l1,m,P) and line_parallel_to_line_through_point(l2,m,P) => l1=l2 (assuming we are working in Euclidean geometry)

MH:

  • line_parallel_to_line_through_point(l,m,P) => (forall Q in Point, point_on_line(Q,l) => distance(Q,m) = distance(P,m)) and a similar one with the role of l and m reversed.

This also begs the question of whether to include all “inherited” properties. For example, this property would follow logically if we added

line_parallel_to_line_through_point(l,m,P) => line_parallel_to_line(l,m) (even though the latter is something different in a DGS construction)

and the latter symbol had the FMPs:

line_parallel_to_line(l,m) => line_parallel_to_line(m,l)

and

line_parallel_to_line(l,m) => (forall P,Q : Point, point_on_line(P,l) => point_on_line(Q,l) => distance(P,m) = distance(Q,m))

Examples

  1. parallel and perpendicular lines through points example: source JSXGraph