Prove that the internal bisectors of the angles of a triangle are concurrent

Show that the perpendicular bisectors of the sides of a triangle are concurrent.

Prove that the internal bisector of the angle A of a triangle ABC divides BC in the ratio AB:AC

Show that the altitudes of a triangle are concurrent

Prove that the medians of a triangle are concurrent.

Prove that the altitudes of a triangle are concurrent.

Prove using vectors: Medians of a triangle are concurrent.

Prove that the bisectors of the interior angles of a rectangle form a square.

Prove that If the bisector of any angle of a triangle and the perpendicular bisector of its opposite side intersect, they will intersect on the circumcircle of the triangle.

Prove that the bisectors of opposite angles of a parallelogram are parallel.

The internal bisectors of the angles of a triangle ABC meet the sides in D, E, and F. Show that the area of the triangle DEF is equal to (2/_\abc)/((b+c)(c+a)(a+b) , where /_\ is the area of ABC.