The commands in this package enable you to work in two-dimensional Euclidean geometry.  Note that the package does not support the extended plane, that is, it does not handle points at infinity and the line at infinity.

Each command in the geometry package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The geometric objects supported in this package are: point, segment, directed segment, line, triangle, square, circle, ellipse, parabola, hyperbola, and conic (including the degenerate cases). To create these geometric objects, use the following commands.

Triangle geometry ranks among the most enduring topics in all of mathematics. The following commands relating to a triangle are supported.

For other geometric objects, the following commands are supported.

Point:

Segment/Directed Segment:

Square:

Line:

Circle:

Ellipse:

Parabola:

Hyperbola:

Polygon:

Various transformations are supported.

Graphics: the draw command provides the graphical visualization of all objects supported in the package.

Other routines: various other commands are also implemented.

To display the help page for a particular geometry command, see Getting Help with a Command in a Package.

When an object is defined through its algebraic representation (an equation or a polynomial), you can use any name for the horizontal axis and vertical axis. In general, the names of the axes must be included when you define an object. A simple way to set the names without being prompted is to set the environment variables _EnvHorizontalName and _EnvVerticalName to the axes names that you prefer; otherwise, Maple will prompt you to input of the name of the axes. In this case, simply types a name and a semicolon (or colon) for each query.

For commands in the package that create a geometric object, or a list of geometric objects, the calling sequence is of the form command_call(obj,...);, where obj is either a name of the geometric object to be created, or a list of geometric objects to be created.

Note that you must make explicit assumptions for the symbolic names in an object (for example, real, positive, ...) when you want to apply a test (for example, IsOnLine) to an object. In this case, the power of this package is dependent on the power of the Maple assume command.

For commands where output is a Boolean value (true, false, FAIL), the calling sequence is of the form command_call(..., cond);, where cond is a an optional name. If the output is FAIL, and this optional argument is given, then the condition that makes the output be true is assigned to cond. cond might be a Maple expression (use assume(cond);), or of the form $\mathrm{cond}=\&\mathrm{or}\left(\mathrm{expr\_1},...,\mathrm{expr\_n}\right)$ or $\mathrm{cond}=\&\mathrm{and}\left(\mathrm{expr\_1},...,\mathrm{expr\_n}\right)$ (use assume(op(i, cond)); for the former case where i is from 1 to n; and assume(op(cond)); for the latter case.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{circle}\left(c\,x^2+y^2=1\,\left[x\,y\right]\,'\mathrm{centername}'=o\right)$

$c$

$\mathrm{detail}\left(c\right)$

$\begin{array}{ll}\text{name of the object}& c\\ \text{form of the object}& \mathrm{circle2d}\\ \text{name of the center}& o\\ \text{coordinates of the center}& \left[0\,0\right]\\ \text{radius of the circle}& 1\\ \text{equation of the circle}& x^2+y^2-1=0\end{array}$

$\mathrm{circle}\left(c\,a^2+b^2=1\right)$

$c$

$\mathrm{detail}\left(c\right)$

$\begin{array}{ll}\text{name of the object}& c\\ \text{form of the object}& \mathrm{circle2d}\\ \text{name of the center}& \mathrm{center\_c}\\ \text{coordinates of the center}& \left[0\,0\right]\\ \text{radius of the circle}& 1\\ \text{equation of the circle}& a^2+b^2-1=0\end{array}$

$\mathrm{\_EnvHorizontalName}≔'m'\:$$\mathrm{\_EnvVerticalName}≔'n'\:$

$\mathrm{circle}\left(c\,\left[\mathrm{point}\left(\mathrm{oo}\,0\,0\right)\,1\right]\right)$

$c$

$\mathrm{Equation}\left(c\right)$

$m^2+n^2-1=0$

$\mathrm{line}\left(\mathrm{l2}\,x+y=1\,\left[x\,y\right]\right),\mathrm{circle}\left(c\,x^2+y^2=1\,\left[x\,y\right]\right)\:$

$\mathrm{intersection}\left(H\,\mathrm{l2}\,c\,\left[M\,N\right]\right)$

$\left[M\,N\right]$

$H$

$\left[M\,N\right]$

$\mathrm{detail}\left(H\right)$

$\left[\begin{array}{ll}\text{name of the object}& M\\ \text{form of the object}& \mathrm{point2d}\\ \text{coordinates of the point}& \left[0\,1\right]\end{array}\,\begin{array}{ll}\text{name of the object}& N\\ \text{form of the object}& \mathrm{point2d}\\ \text{coordinates of the point}& \left[1\,0\right]\end{array}\right]$

$\mathrm{line}\left(\mathrm{l2}\,x+y=1\,\left[x\,y\right]\right),\mathrm{point}\left(A\,a\,\frac{1}{2}\right),\mathrm{point}\left(B\,\frac{3}{5}\,b\right)\:$

$\mathrm{IsOnLine}\left(\left\{A\,B\right\}\,\mathrm{l2}\,'\mathrm{cond}'\right)$

IsOnLine:   "hint: the following conditions must be satisfied: {-2/5+b = 0, -1/2+a = 0}"

$\mathrm{FAIL}$

$\mathrm{cond}$

$\left(-\frac{2}{5}+b=0\right)\&and\left(-\frac{1}{2}+a=0\right)$

$\mathrm{assume}\left(\mathrm{op}\left(\mathrm{cond}\right)\right)$

$\mathrm{IsOnLine}\left(\left\{A\,B\right\}\,\mathrm{l2}\right)$

$\mathrm{true}$