Problems

A convex polygon \(A_1A_2...A_n\) has the following property: if we parallel push all the lines containing the sides of the polygon for a distance \(1\) outside, we will obtain another polygon, similar to the original one with the corresponding parallel sides of the same ratio. Prove that one can inscribe a circle into the original polygon \(A_1A_2...A_n\).

On the first day Robinson Crusoe tied the goat with a single piece of rope by putting one peg into the ground. What shape did the goat graze?