What is Advanced Euclidean Geometry?
The triangle is perhaps the simplest figure studied in elementary geometry, having been well understood since the time of the Greek geometers. However, some of the most fascinating properties of triangles escaped the Greeks, and were not discovered until the Renaissance in Europe. In fact, many of the ideas we will study have not been known until recent years. For example, a classic theorem states that for any triangle, the midpoints of the sides, the feet of the altitudes, and the midpoints of the orthocentral segments all lie on a single circle, whose radius is exactly half the radius of the circumscribed circle. Moreover, this circle is always internally tangent to the inscribed circle of the triangle. Examples like these bring out the enormous depth of the geometry of triangles, and the extent to which completely unexpected relationships exist between seemingly the simplest concepts. Modern triangle geometry has been described as the field in mathematics with the highest number of
