[alogo] 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}.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]

[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.

See Also


Return to Gallery

Produced with EucliDraw©