Optimal constants in normed planes

We define new geometric constants for normed planes, determine their optimal values, and characterize types of planes for which these optimal values are attained. Relations of these constants to several topics, such as areas and distances from points to sides of triangles, are also investigated. We perform some calculations of new and old constants given by trigonometric functions for certain classes of norms, and a conjecture on vectors which are isosceles orthogonal is disproved.