KöMaL 2006 B3879

An escribed circle of a convex polygon is defined as a circle that touches one side of the polygon externally, and also touches the extensions of the two adjacent sides. Let $k$ denote the sum of the areas of the escribed circles of a polygon, and let $t$ denote the total area of the circles drawn over the sides of the polygon as diameters. Assume that the polygon has an inscribed circle, and let $K$ and $T$ denote the perimeter and area of the inscribed circle, respectively. Show that $\frac{k}{t}\le \frac{K}{T}$ .