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In 1953, Victor Thébault conjectured a link between the altitudes of a tetrahedron and the radical center of the four spheres with the centers at the vertices of this tetrahedron and the corresponding tetrahedron altitudes as radii. This conjecture was proved in 2015. In this paper, we propose an analogue of Th\'{e}bault's theorem. We establish a link between the radical center of the four spheres, the insphere, and the Monge point of a tetrahedron.

In this paper, we establish two strengthened versions of Klamkin’s inequality for an n-dimensional simplex in Euclidean space $${E}^n$$ and give some applications.

We establish a new generalization of the Erdos–Mordell inequality by adding more a set of weights to its terms. The same method is used on two other variants of the Erdos–Mordell inequality which are Barrow’s inequality and Dao–Nguyen–Pham’s inequality. Using these generalizations, we derived some strengthened versions of the original Erdos–Mordell inequality and its variations.

In this paper, we give an explicit description for the automorphism groups of finite multitype models in $$\mathbb C^n$$ .

In the first page of the published article, the author name is mistakenly spelled out as Klamlein. The correct version of the author name is given below