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Angle bisector in isosceles trapezoid (Posted on 2021-08-30) Difficulty: 2 of 5
Given ABCD is an isosceles trapezoid with AB=BC=CD.

Prove the bisector of ∠BAD contains C.

No Solution Yet Submitted by Jer    
Rating: 3.0000 (2 votes)

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soln | Comment 1 of 2
The trisosceles trapezoid is a special case of an isosceles trapezoid, which, by one accepted definition, is a trapezoid with bilateral symmetry about the line that bisects both bases.  Thus, its diagonals are of the same length and, again by symmetry, are split into the same two lengths by the crossing. In this way, the two bases are the bases of two isosceles triangles: BOC and AOD, where O is the crossing point. Since opposite angles BOC and AOD are equal, the two isosceles triangles BOC and AOD are similar and angle BCA = angle CAD. Furthermore, since AB=BC, triangle ABC is isosceles, and angle BAC = angle BCA. Therefore angle BAC = angle CAD with AC the bisector.   

QED   

Edited on August 31, 2021, 4:48 am
  Posted by Steven Lord on 2021-08-30 10:08:15

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