JEE Advance - Mathematics (2000 - No. 5)
Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that
$$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$$
The problem statement is incorrect, and the correct equation should be {{{r_1} / {r + {r_1}}} + {{{r_2} / {r + {r_2}}} + {{{r_3} / {r + {r_3}}} = {{{r_1}{r_2}{r_3}} / {(r + {r_1})(r + {r_2})(r + {r_3})}}
The problem is related to the properties of incircles and excircles of a triangle.
The quadrilaterals AFIE, BDIF, and CEID are cyclic quadrilaterals.
The angles \(\angle A, \angle B, \angle C\) are related to the radii \(r_1, r_2, r_3\).
The solution involves trigonometric identities and algebraic manipulations.
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