In the given figure, \(\frac{AX}{BX} = \frac{AY}{YC}\) and \(\angle BXY = \angle CYX\), check that \(\Delta ABC\) is an isosceles triangle.
Proof:
Given that \(\frac{AX}{BX} = \frac{AY}{YC}\)
It implies that \(XY ||BC\), by the converse of basic proportionality theorem.
\(\Rightarrow \angle AXY = \angle AYX\) - - - (i)
Now, \(\angle AXY = \angle\) ( angles) - - - (ii)
\(\angle AYX = \angle\) ( angles) - - - (iii)
From eqn (i), (ii) and (iii), we get
\(\angle ABC = \angle\)
Sides opposite to equal angles are equal.
\(\Rightarrow AB = \)
Therefore, \(\Delta ABC\) is an isosceles triangle.
Hence proved.