ReflectingSegmentEnds

Reflecting segment ends

Let a segment DE glide on side BC of triangle ABC. Reflect its end-points {D,E} on sides {AB,AC} respectively to obtain points {D₁,E₁}. Line D₁E₁ envelopes a parabola tangent to lines {HB,HC} which are the reflexions of BC with respect to lines {AB,AC}.

[1] Line AH is the bisector of angle(BHC). This because in triangle BCH lines {BA,CA} are external bisectors.
[2] Lines {DD₁,EE₁} which are orthogonal to sides {BA,CA} meet at a point F and triangle DEF remains congruent to itself. Its vertex F moves on a fixed line parallel to BC intersecting lines {BH,CH} at {B₁,C₁} respectively.
[3] Draw from {B₁,C₁} parallels to sides {BA,CA}. They intersect on AH at a point A₁ (because of [1]).
[4] F moves on the basis B₁C₁ of the fixed triangle A₁B₁C₁ (similar to ABC). Points {D₁,E₁} are the reflexions of point F with respect to its sides {B₁A₁,C₁A₁}.
[5] The results of ReflexionsOfLine.html apply to triangle A₁B₁C₁ and point F on its side B₁C₁ and prove the claim.

Reflecting segment ends

See Also