On the medians of a triangle in two-point homogeneous spaces

Abstract

We know that the medians of a triangle in both Euclidean, spherical and hyperbolic planes are concurrent. This is not the case for triangles in both real and complex projective planes. In this article we demonstrate that for a triangle in the complex and the quaternion projective spaces any two medians do not intersect unless the triangle is contained in some totally geodesic $$\mathbb {R}P^2$$ or $$\mathbb {C}P^1$$ . It is seen that if a triangle is contained in $$\mathbb {C}P^2$$ and not in a totally geodesic subspace then the complex projective lines containing a vertex and the middle of its opposite side intersect two by two, yielding a triple of points. It is shown that this triple is reduced to a point if and only if our triangle is a null-homotopic triangle in a real projective subspace. However our three points are distinct for a non-null-homotopic triangle in a real projective subspace. Furthermore it is seen that starting with a triangle $$T_{0}$$ the construction leads to a sequence of triangles $$T_{n}$$ , $$n \in \mathbb {N}$$ . Then we prove that if $$T_{0}$$ is equilateral so is $$T_{n}$$ for every n. It turns out that our sequence of equilateral triangles $$(T_{n})$$ converges to a point, unless $$T_{0}$$ is a non-null-homotopic triangle contained in a real projective subplane $$\mathbb {R}P^2$$ . In which case the sequence converges to the unique equilateral triangle contained in a real projective line $$\mathbb {R}P^1$$ . In addition we prove that starting with a non-equilateral non-null-homotopic triangle in $$\mathbb {R}P^2$$ the construction yields a sequence of non-equilateral triangles which also converges to an equilateral triangle in an $$\mathbb {R}P^1$$ . Finally we get in exactly the same way the corresponding results for triangles in the complex and quaternion hyperbolic spaces.