Prove that the line segment joining the midpoints of two opposite sides of the quadrilateral ABCD bisects the line segment joining the midpoints of the diagonals.

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Among other meanings, a midpoint of a line segment is the center of mass of the segment if the segment is considered uniformly heavy. Also, it is the center of mass of the system of two endpoints if they have the same mass and the segment itself is weightless.

This way, imagine that each vertex of a quadrilateral has the same mass, while the sides are weightless. The midpoint `O_1 ` of a side is the center of mass of two vertices, and the midpoint of the opposite side is the center of mass `O_2 ` of the two other vertices. Because of this, the center of mass of all four vertices is the midpoint of the segment `O_1 O_2 .`

From the other hand, the midpoint `P_1 ` of a diagonal is the center of mass of two vertices, and the midpoint of another diagonal is the center of mass `P_2` of the two other vertices. Because of this, the center of mass of all four vertices is the midpoint of the segment `P_1 P_2 .`

This proves that two segments in question have an intersection point. Moreover, this point is the center of mass of the quadrilateral, and it is the midpoint of both these segments.

Recall that the midpoint of `(x_1,y_1) ` and `(x_2,y_2) ` is `((x_1+x_2)/2,(y_1+y_2)/2). ` This way, the centers of each of the two segments in question have the same coordinates, namely the average of x and y coordinates of the vertices.

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