The word construction is pretty well understood in the common language. But we wish to make it a bit more precise, insofar as all constructions begin with a small set of initial points and lines in the plane. Then tools are applied to these, but the tools need to be specified geometrically. Euclid’s Postulates were all formulated in such terms: e.g. Given two points, to draw the circle centered at one, and through the other.

Subsequent existence theorems furnish new tools. For example, once we may assume the existence of the perpendicular bisector (perbis) of a line segment we may use its properties to constuct it. Of course the easiest construction is to let KSEG do it.

## Construction with KSEG

This computer based geometrical drawing program by Ilya Baran has a row of buttons with icons that look a bit like the construction the button can do. Using KSEG will familiarize you with these buttons. And you can always experiment if you don’t like reading Help. Important is to remember that no button does anything until you have chosen the correct initial objects the button uses for its construction.

• Points are created with a right-click.

• Object are chosen or selected with a left click. They turn red.

• You can unchoose by choosing an empty spot in the plane.

• Hold the shift button down to choose more than one object.

• You can create a point tied to an already existing line with a right click on the line.

• A chosen object can be moved as long as you hold down the left-mouse.

When you have chosen a set of objects, the button that is able to do something new with it turns red, or yellow. Red is for a construction. Yellow is for more complicated operations.

For example, create two points, choose them and observe what all you can do with just two points. Draw the segment between them, draw the ray from the first through the second, draw the infinite line through them, and draw a circle.

A more interesting example is to choose a point not on a line. You can drop the perpendicular, and draw the parallel through a point not on a line .

## Construction with Ruler-and-Compass

This is often misunderstood. Historically it means a straight-edge without markings on it, and compass for drawing arcs of circles, given two points: the center and a radial point. You may even remember how to drop a perpendicular from a point to a line from high school. Do you remember how to draw parallels?

Since we have KSEG for accurate constructions, we shall relax this limited set of tools by including two more tools. The gnomon, which is a device for making right angles. You should acquire a transparent 30-60-90 triangle as your gnomon. In a pinch, the corner of a sheet of paper can serve as a gnomon.

And you can use strips of paper to find midpoints (by folding) and copying distances by making marks with you pen rather than messing with a compass.

Thus, to drop that perpendicular, place you gnomon with one side on the line and the other through the point. To find the parallel, drop the perpendicular, and a perpendicular to it at the point off the original line.

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