Curves and Sectioning Angles Nicholas Molbert, Julie Fink, and Tia - - PowerPoint PPT Presentation

Introduction Basic Constructions The Hyperbola The Problem of Trisection

Curves and Sectioning Angles

Nicholas Molbert, Julie Fink, and Tia Burden July 6, 2012

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection

Table of Contents

1

Introduction History Notation

2

Basic Constructions Propositions and Pictures

3

The Hyperbola General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

4

The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions Three Greek problems of antiquity

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry:

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry:

1

Hippocrates (460-380 BC) labels points and lines

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry:

1

Hippocrates (460-380 BC) labels points and lines

2

Hippias (460-399 BC) and the quadratix

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Plato paves the way for planar constructions Three Greek problems of antiquity The groundbreaking discoveries in classical geometry:

1

Hippocrates (460-380 BC) labels points and lines

2

Hippias (460-399 BC) and the quadratix

3

Menaechmus (380-320 BC) and conic sections

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

2

Pappus (early fourth century) used a hyperbola

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

2

Pappus (early fourth century) used a hyperbola

3

Descartes (1596-1650) used the curve y = x2

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

2

Pappus (early fourth century) used a hyperbola

3

Descartes (1596-1650) used the curve y = x2

The link of Francios Viete (1540-1603)

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

2

Pappus (early fourth century) used a hyperbola

3

Descartes (1596-1650) used the curve y = x2

The link of Francios Viete (1540-1603) Wantzel’s proof of impossibility

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

History of the Problem of Trisection

Trisections with extra tools:

1

Archimedes (287-217 BC) used a marked straight edge

2

Nicomedes (280-210 BC) also used a marked straight edge along with a conchoid

The successful trisections:

1

Apolonius (250-175 BC) used conic sections

2

Pappus (early fourth century) used a hyperbola

3

Descartes (1596-1650) used the curve y = x2

The link of Francios Viete (1540-1603) Wantzel’s proof of impossibility From Greek constructions to abstract algebra

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

Notation

← → XY denotes the line through points X and Y .

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

Notation

← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y .

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

Notation

← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C(X, Y ) denotes a circle with center X and radius XY .

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

Notation

← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C(X, Y ) denotes a circle with center X and radius XY . XY denotes the magnitude of segment XY

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection History Notation

Notation

← → XY denotes the line through points X and Y . XY denotes the segment from point X to point Y . C(X, Y ) denotes a circle with center X and radius XY . XY denotes the magnitude of segment XY ∠XYZ denotes the measure or name of an angle, depending

• n the context.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Propositions

Here are the propositions used in our final trisection construction along with pictures demonstrating construction procedures: Rusty Compass Theorem Copying an Angle Bisecting an Angle Parallel Postulate Raising a Perpendicular

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Rusty Compass Theorem Given points A, B, and C, we wish to construct a circle centered at point A with radius equal to BC.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Copying an Angle Given ∠ABC and a line l containing a point D, we can find E on l and a point F such that ∠ABC = ∠EDF.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Bisecting an Angle Given ∠ABC, there is a point D such that ∠ABD ∼ = ∠DBC.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Dropping a Perpendicular Given a line l and a point p not on l, we can construct a line l′ which is perpendicular to l and passes through p.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Parallel Postulate (Playfair) Given a line l and P not on l, we can construct l′ through P and parallel to l.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures

Raising a Perpendicular Given a point p on a line l, you can construct l′ through point p perpendicular to line l.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Propositions and Pictures Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Important Features of Hyperbolas

Important Features are:

1 focus 2 directrix 3 vertex 4 asymptotes 5 transverse

axis

6 conjugate

axis

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Definition of Hyperbolas

The locus of all points P in the plane the difference of whose distances r1 = F1P and r2 = F2P from two fixed points, called foci, is a constant k, with k = r2 − r1.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Definition of Hyperbolas

The locus of all points for which the ratio of distances from one focus to a line (the directrix) is a constant e (the eccentricity), with e > 1. These loci create two distinct branches of the curve.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

From Definitions to Derivation

So how do we tie all of this together?

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

From Definitions to Derivation

So how do we tie all of this together? Derive a curve that satisfies the specific conditions of the problem.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

From Definitions to Derivation

So how do we tie all of this together? Derive a curve that satisfies the specific conditions of the problem. Use this curve to trisect an arbitrary acute angle.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

From Definitions to Derivation

So how do we tie all of this together? Derive a curve that satisfies the specific conditions of the problem. Use this curve to trisect an arbitrary acute angle. Interpret the features of the curve as it fits into the trisection picture.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Process of Deriving Γ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• Nicholas Molbert, Julie Fink, and Tia Burden

Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• = r sin θ
• csc 22θ − 1

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• = r sin θ
• csc 22θ − 1

= r sin θ √ cot 22θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• = r sin θ
• csc 22θ − 1

= r sin θ √ cot 22θ = r sin θ

• cos 2θ

2 sin θ cos θ

• Nicholas Molbert, Julie Fink, and Tia Burden

Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• = r sin θ
• csc 22θ − 1

= r sin θ √ cot 22θ = r sin θ

• cos 2θ

2 sin θ cos θ

• = r(cos 2θ − sin 2θ)

2 cos θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for w

Values from Derivation Triangle β = x + w y = r sin θ x = r cos θ s = r sin θ

sin 2θ

w2 = s2 − y2 w2 = r2 sin 2θ sin 22θ − r2 sin 2θ w =

• r2 sin2 θ
• 1

sin2 2θ − 1

• = r sin θ
• csc 22θ − 1

= r sin θ √ cot 22θ = r sin θ

• cos 2θ

2 sin θ cos θ

• = r(cos 2θ − sin 2θ)

2 cos θ w = r cos θ 2 − r sin 2θ 2 cos θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for β

β = r cos θ + r cos θ 2 − r sin2 θ 2 cos θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for β

β = r cos θ + r cos θ 2 − r sin2 θ 2 cos θ 2r2 cos2 θ + r2 cos2 θ − r2 sin2 θ = 2βr cos θ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for β

β = r cos θ + r cos θ 2 − r sin2 θ 2 cos θ 2r2 cos2 θ + r2 cos2 θ − r2 sin2 θ = 2βr cos θ

• x − β

3 2 − y2 3 = β2 9

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Solving for β

β = r cos θ + r cos θ 2 − r sin2 θ 2 cos θ 2r2 cos2 θ + r2 cos2 θ − r2 sin2 θ = 2βr cos θ

• x − β

3 2 − y2 3 = β2 9 (x − β

3 )2

• β

3

2 − y2

• β

√ 3

2 = 1

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection General Definitions Bridging the Gap from General Definition to Γ The Curve Γ

Picture of Γ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Construction of the Trisection

Trisection Construction Given AB (with A = (0, 0) and B = (1, 0) on the Cartesian plane) and an angle θ, we can construct the trisection of an arbitrary angle using our derived hyperbola, Γ, along with basic constructions previously outlined.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 1 Construct the hyperbola Γ with right branch’s focus at point B.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 2 Using the Rusty Compass Theorem, construct point D at ( 1

2, 0).

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 3 Construct line l perpendicular to AB at point D by dropping a

• perpendicular. Note that line l is the perpendicular bisector of AB.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 4 Bisect angle θ to obtain θ

2.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 5 From the axioms of basic planar constructions, we are able to construct ray l′ with right endpoint on l at an angle θ

2 from l

measured anti-clockwise.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 6 If line l′ does not contain point A, construct l′′ l′ through point A.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 7 If line l′ does contain point A, obtain point O such that ∠AOD = θ

2.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 8 Construct AO.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 9 Reflect ∠AOD about line l such that it creates ∠DOB by using the construction to copy an angle.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 10 Construct OB. Note that AO = OB, so △AOB is isosceles.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 11 Construct C(O, A). Note that C(O, A) contains both points A and B because OA and OB are radii. Also, ∠AOB = θ.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 12 Obtain point P from the intersection of

⌢

AB and Γ.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Step 13 Construct OP, AP, and PB. Note that △APB, obtained in previous step, is the triangle such that 2∠PAB = ∠PBA.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Theorem If ∠PBA = 2∠PAB, then ∠AOP = 2∠POB. Hence, ∠POB trisects ∠AOB. Proof We know that 1

2ψ = φ and

2φ = 1

2α. Solving for φ in

both equations and equating them, we arrive at

1 2ψ = 1 4α. So, 2ψ = α or

ψ = 1

2α. Notice that

α + ψ = ∠AOB, so ∠AOB = 3ψ or ψ = 1

3∠AOB. Therefore, ψ

trisects ∠AOB.

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Geometric Interpretations of Features of Γ

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Investigation of Further Curves

Can we modify this triangle to section angles into however many parts we want just as we have trisected an angle using this triangle?

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles

Introduction Basic Constructions The Hyperbola The Problem of Trisection Construction of Trisection Proof of Trisection Investigation of Further Curves

Investigation of Further Curves

Yes, we can by using this triangle!

Nicholas Molbert, Julie Fink, and Tia Burden Curves and Sectioning Angles