Viviani's theorem and its generalizations

The present master’s thesis deals with Viviani’s theorem valid in an equilateral triangle and stating that the sum of the distances between any interior point and the sides equals the triangle’s altitude i.e. that the sum of the distances is constant. In the paper it is investigated whether the sum of the distances from an interior point to the sides of a nonequilateral triangle also equals any of the triangle’s altitudes or whether there exists any other relation between the sum of the distances and the altitudes. A further investigation refers to a generalisation of the theorem to other polygons and polyhedra. The generalisation concept on chosen examples is shown by the use of various methods. To this end, convex and concave polygons (or polyhedra) are investigated separately. The conclusion gives concrete examples of dealing with the theorem in class and an example of its use in the drawing of diagrams having the form of an equilateral triangle