In this thesis we studied and solved constructive problems where at least one of the given elements is the length of a triangle’s height or a position of the orthocenter. The thesis consists of four chapters. The first chapter provides a list of definitions and theorems used in the second or third part of the thesis. In the second chapter it is spoken about constructions altogether; what it means to construct a figure, which tools do we use and which of the basic operations are doable with the use of these tools. We study the methodics of solving a constructive problem, stages of solving one (analysis, construction, proof and discussion) and solving methods used in the third part of the thesis (the method of sectioning and the algebraic method). Several constructions of chosen algebraic expressions and geometrical places of points have been explicated in detail. Two auxiliary constructions that have been used later on in the thesis are also listed and explained. The third chapter consists of solving metrical constructive problems. The problems are arranged by complexity. Each of them gives carefully explained stages of solving which are substantiated with a list of images constructed using GeoGebra. The fourth chapter includes positional problems with the default position of the orthocenter and problems given the position of the feet of the altitudes. This section also includes stages of solving and constructed images. Definitions and characteristics of a triangle from chapter one have been used to solve a large number of examples which is marked in the text.