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Kepler's laws of planetary motion are acknowledged as highly significant to the construction of universal gravitation. The present study demonstrates different ways to derive the law of equal areas for the Earth by general geometrical and trigonometric methods, which are much simpler than the original derivation depicted by Kepler. The established law of equal area for the Earth was applied to analyze the angular velocity--or the reciprocal of the distance--for the Earth's orbit around the Sun, and can be defined as a periodic function by analyzing the available data, which helps explain the law of ellipses for the Earth., Comment: 9 pages, 3 figures, 4 tables

A simple proof of Thue theorem on Circle Packing is given. The proof is only based on density analysis of Delaunay triangulation for the set of points that are centers of circles in a saturated circle configuration., Comment: 4 pages and 1 figure

In this article, we review the main results of Volume I of Newton's Principia which relates Kepler's law of planets and universal gravitation. In order to clarify the reasoning of Newton, elementary and simple proofs are given to inspire earnest young students. All proofs are based on either the symmetry of spheres or taking great advantage of Kepler's law of area which also reflects the symmetry of centripedal forces., Comment: 22 pages, 9 figures