Theorems in plane geometry

Theorems in plane geometry are propositions or statements that can be proven based on axioms, definitions, and previously established theorems. Plane geometry deals with flat, two-dimensional surfaces and includes the study of points, lines, angles, shapes (such as triangles, quadrilaterals, and circles), and their properties.

Circles are fundamental shapes in geometry, and several important theorems govern their properties and behaviors. Here are some key theorems about circles: 1. **Circumference Theorem**: The circumference \( C \) of a circle is given by the formula: \[ C = 2\pi r \] where \( r \) is the radius of the circle.

Theorems about polygons constitute a significant part of geometry, focusing on the properties, relationships, and characteristics of various types of polygons.

Barbier's theorem is a result in geometry concerning the relationship between the perimeter of a plane figure and the circumference of a circle that has the same area as that figure. Specifically, Barbier's theorem states that for any plane figure, the perimeter of the figure is greater than or equal to the circumference of the circle that has the same area. The equality holds if and only if the figure is a circle.

Hjelmslev's theorem is a result in the field of projective geometry that relates to the properties of conics (i.e., curves defined by quadratic equations) in projective spaces. Specifically, it addresses the conditions under which a conic in one projective plane can be transformed into an equivalent conic in another projective plane.

Holditch's theorem is a result in the field of geometry, specifically in topology related to convex polyhedra. It states that any two convex polyhedra with the same number of vertices, edges, and faces are combinatorially equivalent, meaning they can be transformed into one another through a series of edge-edge and face-face correspondences while preserving the connectivity structure.

The Midpoint Theorem in the context of conics, specifically concerning ellipses, refers to a property related to the midpoints of line segments connecting points on the ellipse. While the term "Midpoint Theorem" can also be associated with other geometrical contexts, such as triangles, in the realm of conics, it is often used to describe certain relationships and properties referring to the midpoints of chords.

The Mohr–Mascheroni theorem is a result in geometry that states that it is possible to construct any length using only a compass, without the need for a straightedge. This theorem is named after the German mathematician Max Mohr and the Italian mathematician Giovanni Mascheroni, who independently proved this result. The theorem can be surprising because traditional geometric constructions often rely on both a compass and a straightedge.