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 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 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, two vertices of a parallelogram are joined to the mid points of two sides. Prove that these lines divides the diagonal in the picture into three equal parts. .

Prove that the Iine joining the midpoints of the two sides of a triangle is parallel to the third side and half of its length.

In the picture below, two lines are drawn from the top vertex of a triangle to the bottom side: Prove that the ratio in which these lines divide the length of the bottom side is equal to the ratio of the area of the three smaller triangles in the picture.

Show that the line segment joining the midpoints of two 'sides of a triangle is parallel to. the third side and is half its length using vectors.

In this picture,the mid pointsof the sides of the large triangle are joined to make a small triangle inside. Calculate the coordinates of the vertices of the large triangle.

Prove that the two triangles shown below have the same area

In this picture the horizontal lines at the top and bottom are parallel. Prove that the yellow and red triangles are of the same area.