Triangle $ABC$ had incenter $I$. Let points $X,Y$ be located on the line segment $AB,AC$ respectively, so that $BX\cdot AB=IB^2, CY\cdot AC=IC^2$. Given that the points $X,I,Y$ lie on a straight line, find the possible values of the measure of angle $A$.
It is a peculiar type question. I can't answer any this type question. I think we can't do this problem using geometry. Somebody please help me. It is a INMO question. Thank you.
It is just angle chasing. The condition $BX\cdot BA=BI^2$ tells you that the circumcircle of $AXI$ is tangent at $I$ to the $BI$-line. Similarly, the circumcircle of $AIY$ is tangent at $I$ to the $CI$-line. Consider the centres of these circles: they both have to lie on the perpendicular bisector of $IA$. Compute the angles in the resulting configuration and translate the collinearity of $X,I,Y$ in a trigonometric equation in $\widehat{A}$. Solve it and deduce $$ \color{red}{\widehat{A}=60^\circ}.$$ This should help you in chasing angles: