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Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
Geometry (from Ancient Greek γεωμετρία (geōmetría) 'land measurement'; from γῆ (gê) 'earth, land' and μέτρον (métron) 'a measure') [1] is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. [2]
This formula is also known as the shoelace formula and is an easy way to solve for the area of a coordinate triangle by substituting the 3 points (x 1,y 1), (x 2,y 2), and (x 3,y 3). The shoelace formula can also be used to find the areas of other polygons when their vertices are known.
Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: [4] The triangle is a spherical one. Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.