JEE Advance - Mathematics (1998 - No. 3)
Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)
The statement is false; the intersection point does not always lie on the line.
Vector methods can easily prove this by expressing the position vectors of the intersection point and midpoints and showing collinearity.
The intersection point only lies on the line if the non-parallel sides are equal in length.
This can only be proven using coordinate geometry, not vector methods.
The problem requires advanced knowledge of projective geometry, which is beyond the scope of typical linear algebra.
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