The orthocenter is defined as the point where the altitudes of a right triangle's three inner angles meet. It is also the vertex of the right angle.
Since two of the sides of a right triangle already sit at right angles to one another, the orthocenter of the right triangle is where those two sides intersect the form a right angle. The altitude of the third angle, the one opposite the hypotenuse, runs through the same intersection point. The orthocenter of an obtuse triangle lays outside the perimeter of the triangle, while the orthocenter of an acute triangle lays inside the triangle.