## Element classes and construction methods

#### for The Geometry Appletversion 2.2

There are eight different element classes: point, line, circle, polygon, sector, plane, sphere, and polyhedron. Only the first five classes are used in plane geometry. All can be used in solid geometry.

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

#### Special notes

1. Any time two points are needed in a construction, one line may be given instead. For instance, if AB has already been defined as a line that connects points A and B, then the triangle ABC can be defined by either of these:
```	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.

#### Index of tables

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

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

Table 8: Element class polyhedron
Constructions: tetrahedron, parallelepiped, prism, pyramid

### Element class point

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 Ai of the polygon A1A2...An with index 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 nth 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

### Element class line

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 nth 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

### Element class circle

Construction method Construction data Description
[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

### Element class polygon

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 nth face of polyhedron A

### Element class sector

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

### Element class plane

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

### Element class sphere

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

### Element class polyhedron

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 Homepage