Assignment:
Consider a triangle in the plane, with angles , a, b , c.
Assume that the radius of its circle is equal to 1.
1) By decomposing the triangle into six right triangles having the incenter as a common vertex, express the area A of the triangle in term of a, b , c ( the answer should be a symmetric expression). Then use the result to show that A can be expressed as a function of the two variables a and b by the formula
A = cot(a/2) + cot(b/2) + tan((a+b)/2)
b) What is the set of possible values for a and b ? Find all the critical points of the function A in this region.
c) By computing the values of A at the critical points and its behavior on the boundary of the region where it is defined, find the maximum and the minimum of A (could you please justify your answer). Describe the shapes of the triangle corresponding to these two situations.
d) Use the second derivative test to confirm the nature of the critical points you found in b).
Provide complete and step by step solution for the question and show calculations and use formulas.