Orthocenters: Dynamic Exploration of Triangle Altitudes

This GeoGebra example dynamically demonstrates the construction of the orthocenter (the intersection of the three altitudes) of a triangle and its relationship to triangle types (acute, right, obtuse, equilateral). The core mathematical concepts include the definition of the orthocenter, the concurrency of altitudes, and classification of triangles by angles. The construction uses commands such as Point to create vertices A, B, C, Polygon to form the triangle, OrthogonalLine to draw altitudes from each vertex to the opposite side, and Intersect to find the orthocenter H. The example includes multiple triangles (Acute, Obtuse, Right, Equilateral) with their respective orthocenters (D, E, F, etc.), likely allowing dynamic manipulation of vertices to observe how the orthocenter moves: inside for acute triangles, at the right-angle vertex for right triangles, and outside for obtuse triangles. This is ideal for middle or high school geometry classes to visualize the orthocenter, understand triangle classification, and explore the property that altitudes are concurrent. It also serves as a hands-on tool for inquiry-based learning.