TED Talk: House of the Future addresses the followingin High School Geometry.
Make formal constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.)
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line distance along a line, and distance around a circular arc.
Prove theorems about lines and angles.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Use coordinates to prove simple theorems algebraically.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.