Given ABCD is an isosceles trapezoid with AB=BC=CD.

Prove the bisector of âˆ BAD contains C.

An isosceles trapezoid is a cyclic quadrilateral and thus has a circumcircle. All four points A,B,C,D are on the circumcircle.

Given segment AB = segment BC = segment CD, then arc AB = arc BC = arc CD.

Arc AB subtends angle ADB, arc BC subtends angles BAC and BDC, and arc CD subtends angle CAD.

Since the arcs are congruent then all four angles ADB, BAC, BDC, and CAD are congruent.

Angles ADB and BDC are then congruent halves of angle ADC with bisector BD. Also, angles BAC and CAD are congruent halves of angle BAD with bisector AC.

Then trivially, it follows that the bisector angle BAD contains point C, and the bisector of angle CDA contains point B.