A Study of the Area of the Inscribed Ellipse of Triangle

Abstract

This research introduces a new method to calculate the area of inellipse of triangle. This formula is derived using the coordinates in the complex plane, the geometrical property of the ellipse, and the Siebeck-Marden theorem rather than the projective transformation and Affine transformation used in the conventional formula derivation. Through this paper, the following results can be obtained. Firstly, we calculated the area of inellipse of triangle related to the ratio of a divided triangle edge determined by the tangency point. Secondly, we derived the area and the ratio of a divided triangle edge of various well-known inellipses of triangle. Thirdly, we reconfirmed that the Steiner inellipse has the maximum area among all inellipses of triangle. This research established the relationship between the area of inellipse of triangle and the Siebeck-Marden theorem, which gives the two foci of inellipse. The new approach could contribute to further research of inellipses.