In the picture below, the side AC of the triangle ABC is extended by D, by adding the length of the side CB. Then the line through C parallel to DB is drawn to meet AB at E.
Prove that CE bisects
`angle C`

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

In the picture below, the top vertex of a triangle is joined to the midpoint of the bottom side of the triangle and then the midpoint of this line is joined to the other two vertices. Prove that the areas of all four triangles obtained thus are equal to a fourth of the area of the whole triangle.

In the picture below, the top vertex of a triangle is joined to the midpoint of the opposite side and then the point dividing this line in the ratio 2:1 is joined to the other two vertices. Prove that the areas of all three triangles in the picture on the right are equal to a third of the area of the whole triangle

In the picture below, the top vertex of a triangle is joined to the midpont of the opposite side and then the point dividing this line in the ratio 2:1 is joined to the other two vertices: Prove that the areas of all three triangles in the picture on the are equal to a third of the area of the whole triangle.

In the parallelogram ABCD , the line drawn through Q point P on AB , parallel to BC meets AC at Q . The line through Q , parallel to AB meets AD R . Prove that (AP)/(PB)=(AR)/(RD) . .

In the figure, squareP=90^o . Sides of triangle APC are extended to B and D. What about the position of P, if the circle is drawn with AD as diameter? .

In the figure, squareP=90^o . Sides of triangle APC are extended to B and D. Prove that, the circles drawn with the sides of a quadrilateral with perpendicular diagonals, will meet at a common point. .