In the triangle ABC, point O is a moving point on the side of AC, and a straight line Mn / / BC is made through point O. let the bisector of the intersection angle BCA of Mn be at point E, and the bisector of the outer angle of the intersection angle BCA be at point E Line and point F. (1) explore the quantitative relationship between line OE and of and prove it. (2) when point O moves on edge AC, will quadrilateral BCFE be a diamond? If so, please prove it. If not, explain the reason. (3) when point O moves to where, and what conditions does triangle ABC meet, quadrangular aecf is a square?

In the triangle ABC, point O is a moving point on the side of AC, and a straight line Mn / / BC is made through point O. let the bisector of the intersection angle BCA of Mn be at point E, and the bisector of the outer angle of the intersection angle BCA be at point E Line and point F. (1) explore the quantitative relationship between line OE and of and prove it. (2) when point O moves on edge AC, will quadrilateral BCFE be a diamond? If so, please prove it. If not, explain the reason. (3) when point O moves to where, and what conditions does triangle ABC meet, quadrangular aecf is a square?

(1) OE = of
CE and CF are angle BCA and bisector of its outer angle respectively. The easy to prove angle is ECF = 180 / 2 = 90 degrees
Angle OEC = angle ECB = angle OCE, similarly, angle OFC = angle OCF, so OE = OC = of
(2) If BCFE is a diamond, then EFC = b = FCC '(Note: C' is any point on the BC extension line)
And the angle ACC '= angle B + angle A. therefore, unless angle a = angle B, BCFE cannot be a diamond (or parallelogram)
(3) If aecf is a square, according to the reasoning in (2), first of all, angle a must be equal to angle B, so that there is CF / / ab
Secondly, the angle ace = angle ACF = 45 degrees, so AC must be perpendicular to BC, that is, ABC must be an isosceles right triangle,
When o moves to the midpoint of AC, aecf is square