### Areas associated with a Strictly Locally Convex Curve

DOI QR Code

Kim, Dong-Soo;Kim, Dong Seo;Kim, Young Ho;Bae, Hyun Seon

• 투고 : 2014.11.07
• 심사 : 2015.04.03
• 발행 : 2016.06.23
• 4 0

#### 초록

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 ${\Delta}ABP$. 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 ${\mathbb{R}}^2$, these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property $S={\frac{4}{3}}T$ for parabolas we study strictly locally convex curves in the plane ${\mathbb{R}}^2$ satisfying $S={\lambda}T+{\nu}U$, where ${\lambda}$ and ${\nu}$ are some functions on the curves. As a result, 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.

#### 키워드

triangle;area;parabola;strictly locally convex curve;plane curvature

#### 참고문헌

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#### 과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)