The ratio of the radius of the circle inscribed in a triangle to the radius of the circle circumscribed around the triangle can vary between what values? When is it the largest?
Let the radius of the inscribed circle of the triangle be $\varrho$, and the radius of the circumscribed circle be $r$. Bisect the sides of the triangle with $s$, and draw a circle around the triangle determined by the midpoints. The radius of this circle is $r / 2$. Now draw tangents to this circle that are parallel to the sides of the original triangle. These tangents form a triangle similar to the original one, which contains the original triangle. Therefore, the inscribed circle of this triangle is at least as large as the original one. But this inscribed circle is our side bisector circle, so $r / 2 \geqq \varrho$, that is, $\varrho / r \leqq 1 / 2$. Equality only holds if the two triangles coincide, that is, if the bisector circle touches all three sides. This only happens in the case of an equilateral triangle.

The radius of the inscribed circle can be any positive value, no matter how small, if the triangle is "almost" degenerate.

Note: The problem has many known solutions.

The solution chosen here originates from István Ádám, who fell victim to the fascist war at a very young age, not quite 20 years old. However, we share this solution not just out of respect. This solution stands out for its simplicity and elegance. The fact that this is the "real" proof of the theorem is best demonstrated by the fact that the problem posed by Lipót Fejér, whether the analogous $R \geq 3 r$ inequality holds for the radii $r$ and $R$ of the sphere inscribed in a tetrahedron and the sphere circumscribed around it, was not transferable to the spatial case before Ádám's solution. However, Ádám's solution can be transferred word for word to the spatial case. Ádám's beautiful result was later significantly generalized by László Fejes Tóth in two excellent papers. For more on this, see Pál Turán: Remembrance. (Mathematical Papers I. 1. issue, pp. 3-15, specifically pp. 13, 14.)