mathcomesalive

Geostruct is a program designed for the investigation and exploration of geometric concepts, ideas, and facts (theorems). It operates as a construction tool, where the user starts off by choosing a number of basic geometric objects (points, lines and circles), makes connections between them (eg put a point onto a line), and then uses them as the basis for geometric actions (eg get the line passing through two chosen points). These new, derived objects can then be used as well in the creation of more objects.

The general approach is that of the standard ruler and compass construction technique, but I have added a number of shortcuts (eg line through a point parallel to another line), a number of special functions dealing with touching and rolling circles, and an optional grid enabling the user to find lengths and to see the algebraic relations describing lines and conics.

The program can be used for very basic geometrical concepts such as:

the difference between point is on a line and line passes through a point,

point midway between two others,

parallel lines,

bisection of an angle,

isosceles and equilateral triangles,

right angled triangles and Pythagoras’ theorem,

general angle properties of triangles

through to

altitudes of a triangle and the orthocentre

angle bisectors and the incentre

perpendicular bisectors of the sides and the circumcentre

intersection of two circles

properties of circles such as angles in the same segment

and on to

tangents to circles, parabolas, ellipses and hyperbolas and tangents

coordinate geometry

touching and rolling circles

There is a collection of working examples, complete with instructions on how to create them, which can then be investigated, manipulated, and modified. There are also some more exotic examples, such as the construction of a parabola from the focus/directrix definition.

The tracking feature lets the user select a point or a line, which, when the display is changed dynamically, leaves a track or trail behind. This can be used for drawing epicyclic curves among other things.

Illustrations

A simple collection of circles, points and one line

An arrangement of touching circles which allows the drawing of an epicyclic curve

Circles touching circles and a circle touching a line. Move point D and the circles will rotate correctly according to their diameters. Move the centre point, I, of the lower circle and it will roll along the red line.

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