Introduction to the Geometry Package

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This worksheet demonstrates some aspects of the geometry package.

Warning: This worksheet is computationally intensive. You may want to close any other memory-intensive applications before working through this worksheet.

Envelope of Circles Method

The following sections show how to use the geometry package to construct some curves based on the method called an envelope of circles. (Reference: Geometry And The Liberal Arts by Dan Pedoe)

Constructing a Cardioid

Define the base circle.

Generate a random point A0 on the circle, which is the cusp of the cardioid.

Generate 50 circles that envelop the constructed cardioid.

n := 50; while i <= n do randpoint(A || i, c0); if evalf(HorizontalCoord(A0)-HorizontalCoord(A || i)) <> 0 then circle(c || i, [A0, A || i]); i := i+1 end if end do

Graph the cardioid curve.

Constructing a Limacon de monsieur Pascal

Two Limacon examples are provided in this section.

Limacon with an inner loop

Define the base circle.

Pick a point outside the circle.

Generate 50 circles that envelop the constructed limacon.

while i <= n do randpoint(A || i, c0); if evalf(HorizontalCoord(A0)-HorizontalCoord(A || i)) <> 0 then circle(c || i, [A0, A || i]); i := i+1 end if end do

Graph this curve.

Limacon without an Inner Loop

Define the base circle.

Pick a point inside the circle c0.

Generate 50 circles that envelop the constructed limacon.

while i <= n do randpoint(A || i, c0); if evalf(HorizontalCoord(A0)-HorizontalCoord(A || i)) <> 0 then circle(c || i, [A0, A || i]); i := i+1 end if end do

The limacon without an inner loop looks like the following:

Constructing a Nephroid

Define the base circle.

Pick a diameter of c0.

Generates 60 points on the circle, and for each point Ai, construct a circle with center Ai that touches the line l.

for i to n do randpoint(A || i, c0); circle(c || i, [A || i, distance(A || i, l)]) end do

The Nephroid can be constructed by using the draw function.

Euclidean Geometry Theorems

The following two sections demonstrate how the geometry package illustrates two theorems in Euclidean geometry. (Reference: A Survey of Geometry by Howard Eves)

Feuerbach's Theorem

Theorem: The nine-point circle of a triangle is tangent to the incircle and to each of the excircles of the triangle.

Define triangle ABC.

Find the incircle inc of ABC.

Find the excircles ex1, ex2, and ex3 of ABC.

Find the nine-point circles ec of ABC.

Check if inc, ex1, ex2, and ex3 are tangent to ec.

AreTangent(ec, inc);

(2.1.1)

Feuerbach's Theorem can now be illustrated.

Nine-Point Circle Theorem

Theorem: In triangle A1A2A3, let M1, M2, M3 be the midpoints of the sides A2A3, A3A1, A1A2, let H1, H2, H3 be the feet of the altitudes on these sides, and let N1, N2, N3 be the midpoints of the segments A1H, A2H, A3H, where H is the orthocenter of the triangle. Then, the nine points M1, M2, M3, H1, H2, H3, N1, N2, N3 lie on a circle whose center N is the midpoint of the segment joining the orthocenter H to the circumcenter O of the triangle, and whose radius is half the circumradius of the triangle.

Define the triangle A1A2A3.

Find the midpoints of A2A3, A1A3, A1A2.

Find the orthocenter and circumcenter of A1A2A3.

Find the altitudes of A1A2A3.

Define the points N1, N2, N3.

Check if M1, M2, M3, H1, H2, H3, N1, N2, N3 are on circle c1.

(2.2.1)

The Nine-Point Circle Theorem can now be plotted.

draw([c1(filled = true, color = ('COLOR')(RGB, 1.0, 1.0, .8)), c(filled = true, color = green), T(style = line, color = blue), N, M1, M2, M3, H1, H2, H3, N1, N2, N3, H], scaling = constrained, style = point, symbol = diamond, axes = none, title = "Nine-Point Circle")

Transformations

The remaining sections illustrate the use of some transformations supported in the geometry package, such as dilatation, homology, rotation, and translation.

Rotation of a Square

for i to n do rotation(t || i, s1, 2*i*Pi/n, counterclockwise, A) end do:

Homology of a Square

for i from 2 to 10 do homology(sq || i, sq || (i-1), o, (1/4)*Pi, counterclockwise, 1/sqrt(2)) end do:

draw([seq(sq || i, i = 1 .. 10), sq(color = COLOR(RGB, 1/2, 1/2, 1/2), filled = true)], color = 'white', thickness = 1, scaling = constrained, axes = none, printtext = false, title = "Homology Transformation of a Square", titlefont = [TIMES, ROMAN, 16])