Angle bissectors of a triangle. Center of the inscribed circle.
Triangle ABC and his 3 angle bissectors. The 3 angle bissectors intersect at a point I. The lines perpendicular to the sides of the triangle originating from point Cic intersect the sides at three points, D, E and F, respectively.
Angle bissector
What's an angle bissector ?
Activity - using Geogebra
Do the bisectors of the angles of a triangle always intersect at a point I ?
Can we always draw an inscribed circle from this point?
Activity :
_ Draw a triangle ABC by placing 3 points randomly.
_ Draw the angle bisectors of the three vertices of the triangle.
_ Place point I at the intersection of the bisectors.
_ Move points A, B, and C to verify that the bisectors always intersect at point I.
_ Draw the three lines perpendicular to the sides of the triangle from point I.
_Place the points of intersection E, D, and F.
_Draw the circle with centre I passing through E.
_ Move points A, B, and C to verify that the centre thus drawn is still a circle inscribed in the triangle.
_ Draw segments [EI], [DI], and [FI].
In the Algebra pane of Geogebra, note the lengths of these segments.
_Move points A, B and C, then note the new values of segments [EI], [DI] and [FI].
_Answer the following questions
Segments [EI], [DI] and [FI] are ...
The angle bissectors are always ...
Can we always draw a circle inscribed in the triangle using the point of intersection of the bisectors ?