Altitude to the Hypotenuse: Geometric Mean via Similar Triangles

This GeoGebra applet lets students construct the altitude from the right angle of a right triangle to its hypotenuse, then visually discover (or recall) the geometric-mean relationship produced by that altitude. The key ideas are the geometric mean theorem for right triangles, similarity of triangles created by the altitude, and the use of angles and perpendicular constructions to ensure the correct configuration. Construction proceeds by creating the base geometry (using a semicircle/Thales-type setup), forming the right triangle with points and segments, drawing the orthogonal altitude, and locating intersections that define the hypotenuse segments. The applet further supports consistent building and placement using geometric transforms such as rotation, translation, and dilation, along with vector representations for directed construction. Commands highlighted in the underlying build include Semicircle, Point, Segment, Min, Midpoint, OrthogonalLine, Circle, Intersect, Angle, Polygon, CircleSector, Dilate, Vector, Translate, CircleArc, and Rotate, with objects like A, B, C, d, e, poly1/poly2, and intersection points (e.g., C_1, d_1, e_1, d_2, e_2) marking the relevant triangle parts. Suitable for secondary-school geometry lessons, exploratory learning, and quick classroom demonstrations of similarity and geometric mean in right triangles.