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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 51M04: Elementary Euclidean geometry (2-dimensional)

Introduction

Ordinary plane geometry (such as is studied in US secondary schools) holds an irresistible appeal, although many results derive what appear to be unimaginative conclusions from tortured premises. Nonetheless, from time to time something catches our eye and gets us to think about ordinary triangles and circles.

History

Applications and related fields

Constructibility with compass and straightedge is dealt with elsewhere.

Tilings and packings in the plane are part of Convex Geometry.

Many topics regarding polygons (e.g. decompositions into triangles and so on) are treated as part of polyhedral geometry.

Subfields

Parent field: 51M - Real and Complex Geometry

Textbooks, reference works, and tutorials

Software and tables

A compendium of plane curves

For computational geometry see 68U05: Computer Graphics

Other web sites with this focus

• The Geometry Junkyard has a "pile" for planar geometry (and other related topics of interest!)

Selected topics at this site

• Surely a FAQ: How can you tell whether two line segments intersect?
• How do you compute the intersection of lines in a plane?
• Ptolémée's theorem on quadrilaterals inscribed in circles
• Thebault's problem (circles and triangles) -- a difficult elementary problem!
• Steiner-Lehmus Theorem on angle bisectors in a triangle
• Determining ellipses tangent to a collection of lines
• A triangle question whose solution depends on its premise of special 'adventitious' angles. (This is an example of trying to forgo the use of trigonometry when its use would be straightforward but inelegant).
• What happens when you trisect the sides of a triangle and look at the intersection of those lines? ("Marion's theorem")
• Morley's theorem about the trisected angles in a triangle.
• References on Morley's theorem.
• Nice proofs of Morley's Theorem
• Ceva's theorem (and Menelaus' theorem) on line segments associated with a triangle.
• The Fermat-Torricelli point in a triangle
• Some questions about the regular nonagon (nine-sided polygon).
• An unusual question regarding p-sided figures where p is prime or pseudo-prime! (e.g. 341)
• How to find the center of an ellipse with Euclidean tools? (Includes Newton's theorem on secants.)
• What is the nearest point on an ellipse from a given point? (Example of Lagrange Multipliers)
• How can you decide whether two ellipses intersect? (long use of analytic geometry and then symbolic algebra).
• Also in the algebra department: a derivation of Heron's formula for the area of a triangle.
• A generalization of Heron's formula to pentagons.
• Pick's theorem (Area of a lattice triangle determined by number of interior lattice points): statement, citations, proofs
• Some calculus: how to locate two circles so that the area of the intersection halves the original area?
• Kiss precise -- the four radii of mutually tangent circles
• Find the lines tangent to a pair of circles in the plane.
• How to compute the area of a collection of circles?
• Perhaps we return the favor to analysis by doing analysis geometrically: a study of ellipses to decide whether some inequalities imply another.
• Pointer to software for the Delaunay triangulation for a set of points in the plane.
• Any planar set of area less than 1 can be translated so as to avoid lattice points.
• Curves and land surveying.
• How many colors to color the plane if different colors are required for points a unit distance apart?
• Radius of inscribed circle in a triangle.
• A tough synthetic geometry problem
• Formulae for regular polygons relating number of sides to lengths of sides, perimeter, area, and radii of inscribed and circumscribed circles.
• A simple curve-straightening transformation of (not necessarily simple or convex) polygons.
• The Pasch axiom to define 2-dimensional geometry.