Chord AB Research Articles

Area properties associated with a convex plane curve

Abstract Archimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle △ A ⁢ B ⁢ P {\bigtriangleup ABP} . Recently, the first two authors have proved that this fact is the characteristic property of parabolas. In this paper, we study strictly locally convex curves in the plane ℝ 2 {{\mathbb{R}}^{2}} . As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

Areas associated with a Strictly Locally Convex Curve

Archimedes showed that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area S of the region bounded by the parabola X and chord AB is four thirds of the area T of triangle . It is well known that the area U formed by three tangents to a parabola is half of the area T of the triangle formed by joining their points of contact. Recently, the first and third authors of the present paper and others proved that among strictly locally convex curves in the plane , these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property $S

Center of Gravity and a Characterization of Parabolas

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord AB on the parabola, let us denote by P the point on the parabola where the tangent is parallel to AB and by V the point where the line through P parallel to the axis of the parabola meets the chord AB. Then the center G of gravity of the section lies on PV called the axis of the parabolic section with PG = 3 PV . In this paper, we study strictly locally convex plane curves satisfying the above center of gravity properties. As a result, we prove that among strictly locally convex plane curves, those properties characterize parabolas.

On some arc length properties in Minkowski planes

In this note we discuss a property in Minkowski planes related to the (intrinsic) arc length. Let A and B be two distinct non diametral points on the unit sphere, H the midpoint of the chord AB and C = H/∣∣H∣∣ : we say that for the (shortest) arc joining A and B the arc length property holds if the greatest length of the two arcs joining respectively A, C and C, B is smaller than half the length of the chord plus the norm of the segment CH. We discuss cases in which the arc length property holds and cases in which fails, in particular we show that this property holds for “small” arcs when the sphere is twice differentiable. Preliminary to the proof of this last result we have established some formulas on arc parametrized twice differentiable spheres, here the main tools used are James orthogonality and the duality mapping: as a further application a “short” proof of Schäffer's theorem on the perimeter of spheres is given.

CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE

Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle <TEX>${\Delta}ABP$</TEX>, where P is the point on the parabola at which the tangent is parallel to the chord AB. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With the aid of this characterization of parabolas, using centroid of triangles associated with a curve we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be a parabola.

ON THE ARCHIMEDEAN CHARACTERIZATION OF PARABOLAS

Archimedes knew that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle <TEX>${\Delta}ABP$</TEX> where P is the point on the parabola at which the tangent is parallel to AB. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.

Implementing the Assessment Standards for School Mathematics: Assessing Justification and Proof in Geometry Classes Taught Using Dynamic Software

Consider the midpoints of all the chords that can be drawn from a given point, say, A, on a circle (see fig. 1). Can anything special be found about these midpoints? Using dynamic geometry software, students can trace the locus of these midpoints by dragging the chord AB from point B. That is, they can use the computer mouse to select and hold point 8 as it is moved around the circle. The computer displays a dynamic chord with a fixed endpoint A and traces the path of the midpoints. The small blue dots shown in figure 2 represent the midpoints of the chords generated as point B is dragged around the circle. Figure 2 suggests that these midpoints lie on a circle. Is this observation true? How can we be sure? When presented with this task, a high school student answered, “It forms … it forms a circle! The midpoints … the midpoints when you move it around form a smaUer circle inside the big circle!” When the student was asked to justify his answer, he said, “I can see it before me, and it does form a circle. I have evidence for it.”

A Method for Determining the Curvature of Natural Forms and its Application to Certain Tail Feathers of the Superb LyrebirdMenura novaehollandiae

SummarySmith, L.H. (1990). A method for determining the curvature of natural forms and its application to certain tail feathers of the Superb Lyrebird Menura novaehollandiae. Emu 90, 231–240.A method for determining the curvature pattern of a two-dimensional curve is described and demonstrated by its application to three types of tail feathers of the Superb Lyrebird. Assuming that the curve is part of a circle, the straight line AB joining the ends A and B of the curve is a chord of that circle. The perpendicular bisector of the chord AB passes through the centre of the circle 0 and meets the curve (circle) at C, and at D, a distance of 2R from C, where R is the radius of the circle. Because DOC is a diameter of a circle, angle DAC (= angle DBC) equals 90 degrees (Euclidean theorem) and AC (= BC = L)/2R = Cos θ, where θ is the angle between AC and DOC. Thus, R = L/2 Cos θ. The paper describes how L and θ are measured. Extension of the method to the shorter elements resulting from the first step enables a curvature pattern for the entire curve (feather) to be determined. The plain median rectrix of a Superb Lyrebird, and the median and lyrate rectrices of a mature male Superb Lyrebird, were all more highly curved at the distal end.

The Metamorphosis of the Butterfly Problem

Editor's note. This article illustrates the diversity of geometric techniques that can be brought to bear on a single problem. The author was prompted to examine his ample collection of historical material when a compilation of varied proofs of the Butterfly problem was offered by Kaidy Tan of Fukien Teacher's University, Foochow, Fukien, China. All of these proofs had appeared in print, and this article outlines many of them, providing their historical roots. One of the hardiest of the hardy perennials in the realm of Euclidean geometry is the problem that was dubbed The Butterfly by some as yet unidentified poetic mathematician who fancied the image of a lepidopterous creature in the configuration of the problem. This appellation made its first appearance as the title of solutions published in the American Mathematical Monthly in February 1944 [1]. The name took hold and has probably contributed to some extent to the recent popularity of the problem. My love affair with The Butterfly began thirty years ago with the publication of the following proposal in School Science and Mathematics: In a circle (0), P is the midpoint of chord AB. Chords RS and TV pass through the point P. RV cuts AP at a point M, and ST cuts PB at point N. Prove by high school geometry that MP equals PN.

Problem in Plane Geometry

In a plane, a point O and a straight line OH drawn through O are given. OH is the bisector of an unknown angle YOX, which it is required to determine by the following conditions: A point I, given by position in the plane of the figure, is connected with the straight line OH by the given angle IOH =θ, and by the distance OI=c, from the point I to the vertex of the angle. This point I is the middle of a chord AB inscribed in the angle YOX, Furthermore the product OA × OB of the distances to the point O of the extremities of this chord is equal to a given quantity K 2 .