Question 42
Let $ABC$ be a non-equilateral triangle, and let $R$ be the radius of its circumcircle. The incircle of $ABC$ has centre $I$ and is tangent to side $CA$ at the point $D$ and to side $CB$ at the point $E$.
Let $A_{1}$ be the point of line $EI$ such that $A_{1}I=R$, with $I$ lying between $A_{1}$ and $E$. Let $B_{1}$ be the point of line $DI$ such that $B_{1}I=R$, with $I$ lying between $B_{1}$ and $D$. Let $P$ be the intersection of the lines $AA_{1}$ and $BB_{1}$.
a. Prove that $P$ lies on the circumcircle of triangle $ABC$.
b. Suppose now in addition that $AB=1$ and that $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.
Let $A_{1}$ be the point of line $EI$ such that $A_{1}I=R$, with $I$ lying between $A_{1}$ and $E$. Let $B_{1}$ be the point of line $DI$ such that $B_{1}I=R$, with $I$ lying between $B_{1}$ and $D$. Let $P$ be the intersection of the lines $AA_{1}$ and $BB_{1}$.
a. Prove that $P$ lies on the circumcircle of triangle $ABC$.
b. Suppose now in addition that $AB=1$ and that $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.