Orthocenter Construction and Comprehensive Polygon Geometry

This example demonstrates how to construct the orthocenter of a triangle (the intersection of its three altitudes) using geometric construction, and further explores it in combination with squares, trapezoids, polynomial functions, complex numbers, and rational numbers. The core mathematical concepts include the definition and properties of the orthocenter, perpendicularity and altitudes, interior angles of polygons, and interactions between function graphs and geometric objects. The construction uses GeoGebra commands such as Segment, OrthogonalLine, Intersect, and Angle, creating points A, B, C and lines f, g, h, i, j, k, l, m, n, along with angle measures α, β, γ for verification. It is suitable for middle school geometry, math competition training, or introductory college geometry courses, helping students intuitively understand the orthocenter theorem and dynamic relationships among multiple geometric elements.