Shortest-Perimeter Inscribed Triangle (Orthic Triangle)

This GeoGebra example demonstrates the construction of the inscribed triangle with the minimum perimeter inside a given acute triangle, known as the orthic triangle. The core mathematical concept is geometric optimization: among all triangles whose vertices lie on the three sides of the original triangle, the one with the smallest perimeter is formed by the feet of the altitudes. The construction uses GeoGebra commands: Polygon to create the original triangle ABC, OrthogonalLine to draw perpendiculars from each vertex to the opposite side, Intersect to find the altitude feet D, E, F, and Polygon again to form the inner triangle DEF. Additional arbitrary inscribed triangles (e.g., G, H, I) are constructed for comparison, and their perimeters are measured to verify that the orthic triangle is indeed minimal. This resource is suitable for high school and university students studying geometric optimization, triangle properties, and dynamic geometry exploration, and is also valuable for math competitions and classical geometry demonstrations.