You are given a triangle represented by three half-planes ( ): + , with 2 + 2 0 for = 1,2,3. You would like to find the inscribed circle of the triangle, i.e., the largest circle contained in the triangle

(a) Show that the problem can be formulated as the following optimisation problem by explaining in detail the decision variables, the objective, and the constraints max0,, s.t. (, , ) , = 1,2,3, where (, , ) = max (,):()^ 2+() ^2^2 + . Hint: think about the representation of a circle

(b) Given a set of values (, , ) with 0, show that the optimisation problem max(,) + s.t. ( ) ^2 + ( ) ^2 ^2 , which is used to compute (, , ), is a convex optimisation problem in and .

(c) Derive the KKT condition for the optimisation problem part (b) and show that (, , ) = + + ^ 2 + ^2

(d) Show that the optimisation problem in part (a) can be reformulated as a linear optimisation problem in its decision variables. Hint: use the results in part (c).

(e) Consider the triangle defined by three constraints: + 2 2, 0, and 0. What is the inscribed circle of the triangle? Hint: find the optimal solution of the corresponding linear optimisation problem derived in part (d)