version 2.2

Each class has several construction methods. The eight tables below briefly explain these construction methods.

param="ABC;polygon;triangle;A,B,C" param="ABC;polygon;triangle;AB,C"

2. Those constructions that can *only* be used in solid geometry are indicated with a dark
blue background color.

3. Optional data elements are indicated in the tables by square brackets. In particular,
[*z*] indicates *z* is an optional integer, and [plane *A*] indicates *A* is an
optional plane. The optional data elements are only used in solid geometry; they should always be
omitted in plane geometry.

3a. When an optional third coordinate [*z*] is not specified, it is taken to be 0 so that the
point lies in the *xy*-plane, that is, the plane of the screen.

3b. When an optional plane such as [plane *A*] is not specified, it is assumed to be the
*xy*-plane.

4. Data elements come in three kinds: (1) integers, (2) points, and (3) other elements. These are indicated in separate lines in the tables. As long as the elements of each kind come in the right order, they will be interpreted properly. Elements of different kinds may be listed as you like.

- Table 1: Element class point
- Constructions: free, midpoint, intersection, first, last, center, lineSlider, circleSlider, circumcenter, vertex, foot, cutoff, extend, parallelogram, similar, perpendicular, proportion, invert, meanProportional, planeSlider, sphereSlider, angleBisector, angleDivider, fixed, lineSegmentSlider
- Table 2: Element class line
- Constructions: connect, angleBisector, angleDivider, foot, chord, bichord, perpendicular, cutoff, extend, parallel, similar, proportion, meanProportional
- Table 3: Element class circle
- Constructions: radius, circumcircle, invert, intersection
- Table 4: Element class polygon
- Constructions: square, triangle, quadrilateral, pentagon, hexagon, equilateralTriangle, parallelogram, regularPolygon, starPolygon, similar, application, octagon, face
- Table 5: Element class sector
- Constructions: sector, arc
- Table 6: Element class plane
- Constructions: 3points, perpendicular, parallel, ambient
- Table 7: Element class sphere
- Construction: radius
- Table 8: Element class polyhedron
- Constructions: tetrahedron, parallelepiped, prism, pyramid

Construction method | Construction data | Description |
---|---|---|

free | integers x, y |
a freely dragable point in the screen plane with initial coordinates (x,y,0) |

midpoint | points A, B |
the midpoint of a line AB |

intersection | points A, B, C, D
[plane E] |
the intersection of two lines AB and CD in the plane E |

points B, C
plane A |
the intersection of the plane A and the line BC | |

first | points A, B |
the first end A of the line AB |

last | points A, B |
the last end B of the line AB |

center | circle A |
the center of the circle A |

sphere A |
the center of the sphere A | |

lineSlider | points A, B
integers x, y,[z] |
a point that slides along a line AB with initial coordinates (x,y,z) |

circleSlider | circle A
integers x, y,[z] |
a point that slides along a circle A with given initial coordinates
(x,y,z) |

circumcenter | points A, B, C
[plane D] |
the center of a circle ABC passing through 3 points A, B, and C in the
plane D |

vertex | polygon A
integer i |
a vertex A of the polygon
_{i}A_{1}A_{2}...A with index _{n}i |

foot | points A, B, C |
the foot of a perpendicular drawn from A to a line BC |

point A
plane B |
the foot of a perpendicular drawn from A to a plane B | |

cutoff | points A, B, C, D |
the point E on a line AB so that AE = CD |

extend | points A, B, C, D |
the point E on a line AB so that BE = CD |

parallelogram | points A, B, C |
the 4th vertex D of a parallelogram ABCD given 3 vertices A, B, and
C |

similar | points A, B, D, E, F
[planes C, G] |
the point H so that triangle ABH in plane C is similar to triangle DEF
in plane G |

perpendicular | points A, B,
[plane C] |
the point D so that AD is equal and perpendicular to AB in plane
C |

points A, B, D, E
[plane C] |
the point F so that AF is perpendicular to AB in plane C and equals
DE | |

points A, C, D
plane B |
the point E on the line perpendicular to plane B passing through A so that
the distance from E to B equals CD | |

proportion | 8 points A, B, C, D, E, F, G, H |
the point I on GH so that AB:CD = EF:GI |

invert | point A
circle B |
the image of a point A inverted in the circle B |

meanProportional | 6 points A, B, C, D, E, F |
the point G on EF so that AB:CD = CD:EG |

planeSlider | plane A
integers x, y, z |
a point that slides on the plane A with initial coordinates (x,y,z) |

sphereSlider | sphere A
integers x, y, z |
a point that slides on the sphere A with initial coordinates (x,y,z) |

angleBisector | points A, B, C
[plane D] |
The point at the intersection of the angle bisector of angle BAC and the line BC in
plane D |

angleDivider | points A, B, C
[plane D]
integer n |
The point E on the line BC so that angle BAE is the n^{th}
part of the angle BAC in plane D |

fixed | integers x, y,[z] |
the fixed point with coordinates (x, y, z) |

lineSegmentSlider | points A, B
integers x, y,[z] |
a point that slides along within the line segment AB with initial coordinates
(x,y,z) |

harmonic | points B, C, D |
the harmonic conjugate of B with respect to C and D |

Construction method | Construction data | Description |
---|---|---|

connect | points A, B |
the line AB connecting two points A and B |

angleBisector | points A, B, C
[plane D] |
the line AE bisecting angle BAC with E on BC in plane
D |

angleDivider | points A, B, C
[plane D]
integer n |
the line AE with E on BC so that BAE is the n^{th} part
of the angle BAC in plane D |

foot | 3 points A, B, C |
the line AD drawn perpendicular to BC in the screen plane |

point A
plane B |
the line AD drawn perpendicular to plane B with the point D lying on
B | |

chord | points A, B
circle C |
the intersection of the line AB in the circle C |

bichord | circles A, B |
the common chord connecting the two intersection points of the circles A and
B |

perpendicular | points A, B
[plane C] |
the line AD equal and perpendicular to AB in plane C |

points A, B, D, E
[plane C] |
the line AF perpendicular to AB in plane C equal to DE | |

point A, C, D
plane B |
the line EF perpendicular to plane B passing through A equal to CD
with E lying on B | |

cutoff | points A, B, C, D |
the line AE equal to CD along the line AB |

extend | points A, B, C, D |
the line BE equal to CD so that A, B, and C are collinear with
B between A and C |

parallel | points A, B, C |
the line AD parallel and equal to BC |

similar | points A, B, D, E, F
[planes C, G] |
the line AH so that triangle ABH in plane C is similar to triangle DEF
in plane G |

proportion | 8 points A, B, C, D, E, F, G, H |
the line GI along GH so that AB:CD = EF:GI |

meanProportional | 6 points A, B, C, D, E, F |
the line EG along EF so that AB:CD = CD:EG |

Construction method | Construction data | Description |
---|---|---|

radius | points A, B
[plane C] |
the circle with center A and radius AB in the plane C |

points A, B, C
[plane D] |
the circle with center A and radius BC in the plane D | |

circumcircle | points A, B, C
[plane D] |
the circle passing through 3 points A, B, and C in the plane
D |

invert | circles A, B |
the image of circle A inverted in circle B |

intersection | spheres A, B |
the intersection of spheres A and B |

Construction method | Construction data | Description |
---|---|---|

square | points A, B
[plane C] |
the square on a side AB in plane C |

triangle | points A, B, C |
the triangle ABC given 3 vertices A, B, and C |

quadrilateral | points A, B, C, D |
the quadrilateral ABCD given 4 vertices A, B, C, and
D |

pentagon | points A, B, C, D, E |
the pentagon given 5 vertices |

hexagon | points A, B, C, D, E, F |
the hexagon given 6 vertices |

equilateralTriangle | points A, B
[plane C] |
the equilateral triangle on a side AB in plane C |

parallelogram | points A, B, C |
the parallelogram ABCD given A, B, and C |

regularPolygon | points A, B
integer n |
the regular polygon on a side AB given the number of vertices n |

starPolygon | points A, B
integers n, d |
the star polygon on a side AB given the number of vertices n and the density
d |

similar | points A, B, D, E, F
[planes C, G] |
the triangle ABH in plane C is similar to triangle DEF in plane
G |

application | polygon A
points B, C, D |
the parallelogram equal to the given polygon A with one side BC and one angle
BCD |

octagon | 8 points A, B, C, D, E, F, G, H |
the octagon given 8 vertices |

face | polyhedron A
integer n |
the n^{th} face of polyhedron A |

Construction method | Construction data | Description |
---|---|---|

sector | points A, B, C
[plane D] |
the sector of a circle in plane D given the center A and two points B and
C on the circumference |

arc | points A, B, C
[plane D] |
the sector of a circle in plane D whose arc passes through the three points A, B and
C |

Construction method | Construction data | Description |
---|---|---|

3points | points A, B, C |
the plane passing through points A, B, and C |

perpendicular | points A, B |
the plane passing through point A and perpendicular to line AB |

parallel | plane A
point B |
the plane passing through point A and parallel to plane B |

ambient | point A |
the ambient plane of point A |

circle A |
the ambient plane of circle A |

Construction method | Construction data | Description |
---|---|---|

radius | points A, B |
the sphere with center A and radius AB |

points A, B, C |
the sphere with center A and radius BC |

Construction method | Construction data | Description |
---|---|---|

tetrahedron | points A, B, C, D |
the tetrahedron given four vertices |

parallelepiped | points A, B, C, D |
the parallelepiped with three edges AB, AC, and AD |

prism | polygon A
points B, C |
the prism with base A and side edges parallel and equal to BC |

pyramid | polygon A
point B |
the pyramid with base A and apex B |

David E. Joyce

Department of Mathematics and Computer Science

Clark University

Worcester, MA 01610

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