Let ABC be a triangle and M an interior point. We denoted by RM the sum of the distances of the point M from the vertices of ABC and by rM the sum of the distances of the point M from the sides. We found the inequalities between RM and rM for M=O,G,I,H that is the pericenter the centroid the incenter and the orthocenter respectively.
Category: Mathematics
The applications of Leibniz’s formula in Geometry.
It is well known from Mechanics that the polar moment of inertia of a system of weighted points is minimum about the centroid. This can be expressed geometrically and very interesting results can be obtained.
A conic classification. Nine-point ellipse.
A classification of a conic through three no colinear given points, according the position of its center. Nine point ellipse.
The maximal inscribed and the minimal circumscribed ellipse for a centrally symmetric convex figure.
In this note we study the maximal inscribed and the minimal circumscribed ellipse of a cenrally symmetrc convex figure F and we prove two theorems between the area and the remarcable elements of the figure.
Convex figures with conjugate diameters.
The convex figure K has conjugate diameters if and only if its symmetroid K* is an affine image of a Radon curve
The triangle of minimum perimeter circumscribed to a smooth closed compact convex curve (c) in the plane.
Theorem
For the triangle T of a minimum perimeter circumscribed to a convex figure (c) the excircles of T are tangent to (c).
Van den Berg’s Theorem
Let Q(z)=z3+a1z2+a2z+a3=0 be a cybic in C and z1,z2,z3 the roots denoted in the plane by the points A,B,C. The Steiner ellipse in the triangle ABC is denoted by E and F1, F2 the foci. Van den Berg’s theorem asserts that the roots of the derivate Q'(z) are the complex numbers defined in C by the points F1 and F2.
A characteristic property of the ellipse.
Let (c) be a compact smooth convex curve in E2 and A,B are interior points. The point M is on (c) and we denote e(M) the support line of the point M and e0 the unit tangent vector at the point M. We suppose that: angle (AM,e0)=angle(BM,-e0).
A characteristic property of the ellipse.
Let (k) be a rotund closed convex curve and A,B interior points. We consider, from an external point M the two support lines MM1,MM2 and we assume that for every point M, the angles AMM1 and BMM2 are equal. In this note we answer to the question whether (k) is an ellipse.
A characteristic property of a convex centrosymmetric curve
Theorem: If a closed smooth convex curve (c) in E2 and an interior point O have the property that (c) possesses parallel supporting lines at the endpoints of every chord through O, then the (c) is centrosymmetric and O is the center.