1. In a given triangle, the radius of the inscribed circle and the radii of the excircles form (in some order) consecutive terms of a geometric progression. Determine the largest angle of the triangle.
1. Let $a, b$, and $c$ be the sides of the triangle, $s$ its semi-perimeter, and $r, r_{a}, r_{b}$, and $r_{c}$ the observed radii. From the expressions for the area $P=r s=r_{a}(s-a)=$ $r_{b}(s-b)=r_{c}(s-c)$, we conclude that the quantities $s, s-a, s-b$, and $s-c$ also form a geometric progression. Assuming (without loss of generality) $a<b<c$, we have $s(s-c)=(s-a)(s-b)$, which further simplifies to $s a+s b-s c=a b$, i.e., $(a+b+c)(a+b-c)=2 a b$, and from here we get $a^{2}+b^{2}-c^{2}=0$, so the largest angle in the observed triangle is $90^{\circ}$.