]> Some basic geometry

# Some basic geometry

The context of classical geometry is a domain in which the notions of straight line and parallel do not jar with common intuitions. Some interesting contexts do not follow rules compatible with those intuitions, but a broad class of real-world situations are compatible with them to within close enough tolerances that many honest folk don't care about the difference, in practice. Contexts in which these intuitions are exactly correct are formally described as flat or Euclidean, so I am effectively describing Euclidean geometry. I'll leave it to other pages to address matters more formally, in terms which either address deviations from flatness or formally specify flatness, but I'll here endeavour to set forth some of the more prominent results of those intuitions and some of the consequences that follow for any context in which they hold good enough that you don't care about the difference. Many of the intuitive truths here addressed remain true even in non-Euclidean (or curved) geometries, at least if we add some qualifying clauses, typically confining the points, lines and regions under discussion to some sufficiently small portion of the whole context of the curved geometry.

I'll aim to describe geometries verbally, and I'll also use Scalable Vector Graphics (the W3C specification for an XML-based image format) for diagrams: some browsers shall need a plug-in to help them view these, but those that keep up with the times do now support it. Some implementations are still buggy in places: your mileage may vary !

## Conventional markings

When two lines meet at right angles, it is conventional to mark one of the angles they make with a little box symbol. Other angles are marked with arcs of circles, centred on their vertices. When the direction of an angle matters, it is usual to mark one end of it with an arrow-head indicating the direction; by convention, clock-wise angles are negative and anticlockwise ones are positive. Using the same style (orthodoxly, an angle's arc would be styled by using several arcs of almost equal radius; but I'll prefer to use non-black colours for my styles) for several angle-marker arcs is commonly used to indicate that the angles marked with these arcs are equal. Parallel lines are orthodoxly marked with arrows; if several sets of parallel lines are present in a diagram, one set is marked with single arrows, another with double arrows and so on; but I'll simply give the lines non-black colours. Where some line segments are of equal length, it is common to mark the middle of each with a short crossing line; if further lines equal one another in length, but are not known to be equal to the former, they are likewise marked with two (or more) such crossing lines; however, I'll use a single crossing line in all cases, with non-black colours the same on edges of equal length.

## Polygon

A polygon is a piecewise straight closed loop – that is, a sequence of straight line segments, each starting where the previous ended, with the first starting where the last ends. Normally, we restrict attention to polygons which don't intersect themselves: each of these encloses a single simply-connected region known as its interior; the region around it is known as the exterior. The straight lines making up the polygon are known as its edges: each start or end point of an edge is known as a vertex of the polygon. At each vertex, there is an interior angle between the two edges meeting there, on the side of the interior region; there is also an exterior angle between one edge's continuation past the vertex and the other edge. The interior angle is always positive but the exterior angle may be negative if the extension of an edge, used in constructing it, intrudes into the polygon's interior. Taking due account of the sign of the exterior angle, the sum of exterior and interior angles is always a half turn.

## Constructions

There is a classical scheme of construction using compass and straight-edge; I should enumerate how, using only these, to:

• Bisect a line or an angle
• Subdivide a line into any whole number of equal parts
• Construct an equilateral triangle, a square and a regular hexagon
• Construct a pentagon.

### Diagonal

Suppose you're given a circle, but don't know where its centre is. Draw in a chord to the circle; ideally, its end-points should be a quarter-turn apart around the circle, or thereabouts (between 60 and 120 degrees would be fine, by my guess). Construct perpendicular lines through it at the points where it meets the circle; note where each meets the circle again. This gives us four points, forming a rectangle; construct the two diagonals. Each of these is a diameter of the circle: they meet at the centre, bisecting one another, hence incidentally identifying the radius of the circle.

### TODO

Things I should illustrate and/or prove:

• Apollonius's theorem: (x+y)**2 + (x−y)**2 = 2.(x**2 + y**2)
• Connect the two ends of a chord to the centre of the circle; also to some point on the circumference (other than the chord's end-points); the angle facing the chord at this point on the circumference is half the angle on the corresponding side of the centre.
• In particular, the lines from the two ends of a diameter to a given point, off the diameter, on the circumference of a circle meet in a right angle.
• Draw a line from a point on a circle to meet a diameter at right angles; the length of this line is the geometric mean of the two parts into which it cuts the diameter.
Valid CSS ? Valid XHTML ? Written by Eddy.