Geometric Construction of the Orthocenter of a Triangle

This applet demonstrates how to construct the orthocenter of a triangle using GeoGebra. Given a triangle with vertices A, B, and C, the construction begins by drawing altitudes from each vertex perpendicular to the opposite side. The OrthogonalLine command is applied three times: one line is constructed through vertex A perpendicular to side BC, another through vertex B perpendicular to side AC, and a third through vertex C perpendicular to side AB. These three lines are the altitudes of the triangle. The Intersect command is then used to find the point where these altitudes meet. Regardless of whether the triangle is acute, right, or obtuse, all three altitudes are concurrent, and their common intersection point is defined as the orthocenter, labeled here as D. The applet provides an interactive environment where students can drag the vertices A, B, and C to reshape the triangle in real time and observe how the altitudes and the orthocenter adjust accordingly. This dynamic manipulation allows learners to explore how the position of the orthocenter changes with the triangle. In an acute triangle, the orthocenter lies inside the triangle. In a right triangle, it coincides with the vertex at the right angle. In an obtuse triangle, it falls outside the triangle. These behaviors become immediately visible through interaction, reinforcing the geometric properties of the orthocenter. The applet serves as a visual and hands-on tool for students studying Euclidean geometry, particularly topics related to triangle centers and concurrency. It helps bridge the gap between abstract definitions and concrete visualization, making the concept of the orthocenter more accessible. By engaging directly with the construction, students develop a deeper understanding of altitude relationships, perpendicularity, and the elegant fact that three seemingly independent lines always converge at a single point. This applet is well suited for classroom demonstration, self-guided exploration, or assignment use in geometry courses.