JEE Advance - Mathematics (2000 - No. 9)
Let $$ABC$$ and $$PQR$$ be any two triangles in the same plane. Assume that the prependiculars from the points $$A, B, C$$ to the sides $$QR, RP, PQ$$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $$P, Q, R $$ to $$BC,$$ $$CA$$, $$AB$$ respectively are also concurrent.
The first set of perpendiculars being concurrent implies a certain vector relationship between the vertices of the triangles.
The vector relationship derived from the first concurrency condition can be manipulated to show the concurrency of the second set of perpendiculars.
The concurrency of perpendiculars can be expressed as a determinant condition involving dot products of vectors.
Desargue's theorem provides a more straightforward approach to proving the statement.
This problem has no solution. There can't be concurrency on the second set of perpendiculars.