Geometric Constructions
for Love and Money
Kevin Mirus
Madison Area Technical College
Wisconsin Mathematics Council
36th Annual Green Lake Conference
Friday May 7, 2004
kmirus@matcmadison.edu
http://www.matcmadison.edu/is/as/math/kmirus/
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Some types of geometric construction (Ref. 1):
1. Euclidian (Ruler and Compass) constructions use a ruler with no markings and
a compass which can only be used to draw a circle with a given center point
through another given point. Angles cannot be trisected under these restrictions.
2. Mohr – Mascheroni (Compass Only) constructions use a compass alone;
anything that can be constructed with a ruler and compass can be constructed with
a compass alone (!).
3. Ruler constructions use a ruler alone; anything that can be constructed with a
ruler and compass can also be constructed with just a ruler and four given vertices
of a trapezoid (!).
4. Ruler and Dividers constructions use a ruler and a divider to carry distance.
This allows some constructions that a ruler and compass don’t.
5. Poncelet-Steiner and Double Ruler constructions use either a ruler and a circle
with a known center or a ruler with two parallel edges. All ruler and compass
constructions can be constructed under these circumstances.
6. Ruler and Rusty Compass constructions use a ruler and a compass of fixed
opening; anything that can be constructed with a ruler and compass can be
constructed with a compass alone (!).
7. Stick constructions use sticks of equal length that can be laid on top of each
other. You can do more constructions with sticks than with a ruler and compass.
8. Marked Ruler constructions use a ruler with two marks on its edge. You can do
more constructions with a marked ruler than with a ruler and compass.
9. Paperfolding constructions use folds in paper. Any constructions of plane
Euclidian geometry can be executed by means of creases.
10. Constructions using other instruments like a “tomahawk”, carpenter’s squares,
strings, etc. allow for constructions not possible using a ruler and compass.
11. Approximate constructions are useful when close is good enough…
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Problem: Construct a line segment congruent to a given segment.
Step 2: Use a compass to
measure the distance AB,
then draw an arc centered at
C with radius AB that
intersects l at point D.
Comments: This is not a strict Euclidian construction, since the compass was used to
transfer a distance. This construction can also be used to construct the sum and
difference of two given segments.
Given: Line segment AB
Step 1: Choose any point C
on line l.
Problem: Construct a line segment congruent to a given segment by folding
paper.
Given: A line segment.
Step 1: Fold the given line
segment onto the other line.
Step 2: Crease the paper at
either endpoint of the given
segment. The intersection
of the crease and line are
congruent.
Comments:
Problem: Construct a circle whose radius is equal to the length of a given
segment.
Given: Line segment PQ , point R

Step 1: Draw ray PR .
Continued…
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Step 2: Draw an arc centered at P through Q

that intersects PR . Label the intersection
point A.

Step 3: Draw arcs centered at points Q and A
through point P. Label their intersection X.
Step 5: Draw an arc centered at P through R
that intersects PQ . Label the intersection
point B.
Step 4: Draw ray QX .
Step 6: Draw arcs centered at points R and B
through point P. Label their intersection Y.

Step 7: Draw ray RY . Label the intersection


of RY and QX point S.
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Step 7: Draw the circle centered at R through S. RS = PQ, since PQRS is a parallelogram.
Thus, any radius of this circle is equal to PQ.
Comments: This is a strict Euclidian construction for congruent segments. Its importance is to
show that a Euclidian compass is equivalent to a modern compass (i.e., it's okay to use a
compass to remember distances). Ref. 1.
Problem: Construct an angle congruent to a given angle.
Given: Angle BAC
Step 1: Choose any point D on line l.
Continued…
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Step 2: Draw an arc of any radius centered
at A. Label the points of intersection P and
Q. Create the same radius arc centered at
D. Label the point of intersection E.
Step 3: Draw an arc centered at P through
Q. Draw an arc centered at E with the
same radius. Label the intersection point
F.

Step 4: Draw ray DF . BEDF is congruent to BAC
Comments: This construction can also be used to construct the sum and difference of
angles. A good exercise is to construct a 105 angle using ruler and compass…
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Problem: Construct an angle congruent to a given angle by folding paper.
Given: An angle.
Step 1: Fold the given
angle onto the other line.
Step 2: Crease the paper on
the other edge of the angle.
The crease and line form a
congruent angle.
Comments:
Problem: Construct the perpendicular bisector of a given line segment.
Step 1: Draw arcs centered Step 2: Draw segment PQ .
on A and B that have the
Label its intersection with
same radius greater than the
AB as point M.
distance to the midpoint.
Label the intersections P
and Q.
Comments: Constructing the perpendicular bisector gives you the midpoint for free!
Given: Line segment AB
Problem: Construct a perpendicular bisector by folding paper.
Given: A segment.
Step 1: Create the crease
between the endpoints.
Comments:
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Step 2: Fold one endpoint
onto the other. This crease
is the perpendicular
bisector.
Problem: Construct the bisector of a given angle.
Given: Angle BAC
Step 1: Draw an arc of any radius centered
at A. Label the points of intersection P and
Q.
Step 2: Draw arcs of equal radius from P and Q. The line from A through their
intersection bisects BAC .
Comments:
Problem: Construct the bisector of a given angle by folding paper.
Given: An angle.
Step 1: Fold one edge of
the angle onto the other
edge.
Comments: Cool, huh?
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Step 2: The crease bisects
the angle.
Problem: Trisect a given angle using Pappus' Theorem / Method of
Archimedes.
Given: Angle ABC
Steps: Construct perpendiculars and parallels through A.
Draw an arc centered at A that passes through B in the

interior of the circle. Draw ray BS such that MS = RM =
AB.
Comments: This is a marked ruler construction…
Problem: Trisect a given angle using Archimedes' method.
Given: Angle ABC
Steps: Draw a circle centered at B that passes through A

and C. Draw line AQ such that PQ = BC = AB = PB.
Comments: This is another marked ruler construction…
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Problem: Trisect a given angle by folding paper.

Given: Angle PQR . Call PQ line l, and

QR line m.
Step 2: Construct the perpendicular to line m
through point M by folding line m onto itself
such that the crease contains point M. Call the
crease that is made line b.

Step 1: Locate the midpoint M of PQ by
folding line l onto itself so that point P lands on
top of point Q. Call the crease that is made line
a.
Step 3: Construct the perpendicular to line b
through point M by folding line b onto itself
such that the crease contains point M. Call the
crease that is made line c. Notice that line c is
parallel to line m.
Continued…
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Step 4: Fold the paper so that point P lands on
Step 5: Fold the paper so that the crease
line b and point Q lands on line c. This is tricky contains points Q and Q’. Call the crease line e.
and will take some sliding around to get right.
Line e trisects PQR .
On the diagram above, point P landed on top of
line b at point P’, while point Q landed on line c
at point Q’. Call the crease that is made line d.
Comments: You can do more with paperfolding than with a ruler and compass!
Problem: Trisect a given angle using the Quadratix of Hippias.
Given / Step 1: The quadratix of Hippias is a
 y 
curve defined by the equation x  y cot   .
 2
Draw the curve and lay the angle to be trisected
with its vertex at the origin and one edge on the
x axis.
Continued…
Step 2: Mark the point where the angle
intersects the quadratix curve. Construct a line
parallel to the x axis through this point.
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Step 3: Take the segment on the y axis from the Step 4: Draw the line connecting the origin and
origin to the intersection of the horizontal line
the point of intersection between the previous
and divide it into three pieces using a geometric horizontal line and the quadratix. This line
construction. Construct a line parallel to x axis
trisects the original angle.
through the bottom third.
Comments: The quadratix curve can also be constructed “dynamically”…
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Problem: Trisect a given angle using a tomahawk (hatchet).
Comments:
Problem: Divide a given angle into any odd number of equal angles using
hinged tomahawks.
Comments: s
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Problem: Construct a line perpendicular to a given line and containing a
point not on the given line.
Comments:
Problem: Construct a line perpendicular to a given line and containing a
point not on the given line by folding paper.
Fold the line onto itself and form a crease that contains the point.
Comments:
Problem: Construct a line perpendicular to a given line and containing a
point on the given line.
Comments:
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Problem: Construct a line perpendicular to a given line and containing a
point on the given line by folding paper.
Fold the line onto itself and form a crease that contains the point.
Comments: s
Problem: Construct an equilateral triangle.
Comments:
Problem: Construct an equilateral triangle by folding paper.
Make a square, place a crease up the middle, then fold a vertex of the square onto the
middle crease.
Comments:
Problem: Construct an equilateral using the vesica pisces.
Comments: Things are always a little more interesting with subversion and religion
mixed in…
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Problem: Construct a line parallel to a given line through a point not on the
given line.
Comments: s
Problem: Construct a line parallel to a given line through a point not on the
given line.
Comments: s
Problem: Construct a line parallel to a given line through a point not on the
given line using the “draftsman's cheat”.
Comments:
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Problem: Construct a line parallel to a given line through a point not on the
given line by paper folding.
Construct two perpendiculars…
Comments:
Problem: Partition a given line segment into a given number of congruent
segments.
Comments: This is handy in carpentry for laying out stairs…
Problem: Locate the center of a given circle.
Comments:
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Problem: Construct the tangent line to a circle at a given point on the circle.
Draw a radius to the point, then construct a perpendicular to the radius.
Comments:
Problem: Construct the circumscribed circle of a given triangle.
Comments:
Problem: Construct the inscribed circle of a given triangle.
Comments:
Problem: Construct the "center of gravity" of a given triangle.
Comments:
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Problem: Construct the two tangents to a given circle through a given point
outside the circle.
Comments:
Problem: Construct a circle given its diameter using a framing square.
Use the fact that an angle inscribed across a diameter is 90.
Comments: This works better than a nail, string, and pencil, I’m told.
Problem: Construct a square.
Comments: You can now construct an octagon, 16-gon, etc.
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Problem: Construct a regular pentagon.
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
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Step 7
Comments: s
Problem: Construct a regular hexagon.
Comments: This forms the basis for many a church window. Extending the pattern
makes a pretty tiling.
Problem: Construct a regular hexagon using two vesica pisces.
Comments:
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Problem: Construct a triangle congruent to a given triangle.
Comments:
Problem: Construct a regular pentagon using a framing square.
Comments:
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Problem: Construct an approximation to any regular polygon.
Comments:
Problem: Construct a wedge of a 14-gon with seven sticks.
Comments:
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Problem: Construct a Golden Rectangle.
Comments: This is very useful in designing furniture with nice proportions.
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Other Topics: Tangent Circles
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Other Topics: Spirals
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References
1. Isaacs, I. Martin, Geometry for College Students, Brooks/Cole Thomson
Learning, 0-534-35179-4, 2001.
2. Posamentier, Alfred S. and William Wernick, Advanced Geometric
Constructions, Dale Seymour Publications, 0-86651-429-5, 1988.
3. Martin, George E., Geometric Constructions, Springer, 0-387-98276-0, 1998.
4. Sykes, Mabel, Source Book of Problems for Geometry, Dale Seymour
Publications, 0-86651-795-2, 1912.
5. "Four Ways to Construct a Golden Rectangle," Fine Woodworking, Taunton
Press, February 2004, No. 168, p. 50.
6. Lundy, Miranda, Sacred Geometry, Wooden Books, 0-965-20578-9, 1998.
7. Bedford, John R., Graphic Engineering Geometry, Gulf Publishing Company, 087201-325-1, 1974.
8. "Details on Involute Gear Profiles," How Stuff Works web site (has a great
animation of gears meshing), http://science.howstuffworks.com/gear7.htm
9. "Origami and Geometric Constructions,"
http://www.merrimack.edu/~thull/omfiles/geoconst.html
10. Machinery's Handbook (any edition)
11. "Logarithmic Spiral," mathworld website,
http://mathworld.wolfram.com/LogarithmicSpiral.html
12. Wingeom for Windows, website with freeware for geometric constructions (used
to create all diagrams in this packet)
http://math.exeter.edu/rparris/wingeom.html
13. “Tangent Circles,” a web site: http://whistleralley.com/tangents/tangents.htm
14. “Constructing the Heptadecagon,” a web site:
http://www.mathpages.com/home/kmath487.htm
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