PAVIATH INTEGRATED SOLUTION
PAVIATH INTEGRATED SOLUTION
Paviath
INSTITUTION CAMPUS TRAINING MECHANICAL PHASE III MATHEMATICAL TECHNOLOGY

PHASE III MATHEMATICAL TECHNOLOGY

MATHEMATICAL TECHNOLOGY
COURSE I
101 SYMBOLIC
CALCULUSE
101 CONICS
ALGEBRA GEOMETRY
GEOMETRY PROOF
HIGH SCHOOL
COURSE I
 COURSE BY MATHEMATICAL TECHNOLOGY ON APPLICATION

101 SYMBOLIC GEOMETRY EXAMPLES WITH GEOMETRY EXPRESSIONS Philip Todd

CALCULUS EXPLORATIONS WITH GEOMETRY EXPRESSIONS HANNAH IRINA LYUBLINSKAYA/ HANNAH VALERIY RYZHIK

101 CONIC SECTIONS EXAMPLES WITH GEOMETRY EXPRESSIONS Philip Todd
CONNECTING ALGEBRA AND GEOMETRY THROUGH TECHNOLOGY JIM WIECHMANN

DEVELOPING GEOMETRY PROOFS WITH GEOMETRY EXPRESSIONS™IRINA LYUBLINSKAYA, VALERIY RYZHIK, DAN FUNSCH

EXPLORING WITH GEOMETRY EXPRESSIONS IN HIGH SCHOOL MATHEMATICSIAN SHEPPARD

101 SYMBOLIC

101 SYMBOLIC GEOMETRY EXAMPLES WITH GEOMETRY EXPRESSIONS(Philip Todd)

INTRODUCTION 
EXAMPLE 1: MEDIAN & ANGLE BISECTOR OF A RIGHT TRIANGLEI 
EXAMPLE 2: ANGLES AND CIRCLES 
EXAMPLE 3: RECTANGLE CIRCUMSCRIBING AN EQUILATERAL TRIANGLE 
EXAMPLE 4: AREA OF A HEXAGON BOUNDED BY TRIANGLE SIDE TRISECTORS 
INCIRCLES / CIRCUMCIRCLES / EXCIRCLES / AREAS 
EXAMPLE 5: CIRCUMCIRCLE RADIUS 
EXAMPLE 6: INCIRCLE RADIUS 
EXAMPLE 7: INCIRCLE CENTER IN BARYCENTRIC COORDINATES 
EXAMPLE 8: HOW DOES THE POINT OF CONTACT WITH THE INCIRCLE SPLIT A LINE 
EXAMPLE 9: EXCIRCLES 
NAPOLEON’S THEOREM / PYTHAGORAS DIAGRAM 
EXAMPLE 10: NAPOLEON’S THEOREM 
EXAMPLE 11: AN UNEXPECTED TRIANGLE FROM A PYTHAGORAS-LIKE DIAGRAM 
EXAMPLE 12: A PENEQUILATERAL TRIANGLE 
EXAMPLE 13: ANOTHER PENEQUILATERAL TRIANGLE 
EXAMPLE 14: VON ABUEL’S THEOREM 
CIRCLE COMMON TANGENTS 
EXAMPLE 15: LOCATION OF INTERSECTION OF COMMON TANGENTS 
EXAMPLE 16: CYCLIC TRAPEZIUM DEFINED BY COMMON TANGENTS 
EXAMPLE 17: TRIANGLE FORMED BY THE INTERSECTION OF THE INTERIOR COMMON TANGENTS OF THREE CIRCLES 
EXAMPLE 18: LOCUS OF CENTERS OF COMMON TANGENTS TO TWO CIRCLES 
EXAMPLE 19: LENGTH OF THE COMMON TANGENT TO TWO TANGENTIAL CIRCLES 
EXAMPLE 20: TANGENTS TO THE RADICAL AXIS OF A PAIR OF CIRCLES 
EXAMPLE 21: THE EYEBALL THEOREM 
EXAMPLE 22: A LIMIT POINT ARBELOS 
EXAMPLE 23: TWO CIRCLES INSIDE A CIRCLE TWICE THE RADIUS, THEN A THIRD 
EXAMPLE 24: A THEOREM OLD IN PAPPUS’ TIME 
EXAMPLE 25: ANOTHER FAMILY OF CIRCLES 
EXAMPLE 26: ARCHIMEDES TWINS 
EXAMPLE 27: BUEHLER’S CIRCLE
EXAMPLE 28: CIRCLE TO TWO CIRCLES ON ORTHOGONAL RADII OF A THIRD 
CONICS 
EXAMPLE 29: CIRCLE OF APOLLONIUS 
EXAMPLE 30: A CIRCLE INSIDE A CIRCLE 
EXAMPLE 31: PARABOLA AS LOCUS OF POINTS EQUIDISTANT BETWEEN A POINT AND A LINE 
EXAMPLE 32: PARABOLIC MIRROR 
EXAMPLE 33: SQUEEZING A CIRCLE BETWEEN TWO CIRCLES 
EXAMPLE 34: ELLIPSE AS A LOCUS 
EXAMPLE 35: ARCHIMEDES TRAMMEL 
EXAMPLE 36: AN ALTERNATIVE ELLIPSE CONSTRUCTION 
EXAMPLE 37: “BENT STRAW” ELLIPSE CONSTRUCTION 
EXAMPLE 38: ANOTHER ELLIPSE 
EXAMPLE 39: SIMILAR CONSTRUCTION FOR A HYPERBOLA 
EXAMPLE 40: ELLIPSE AS ENVELOPE OF CIRCLES 
EXAMPLE 41: HYPERBOLA AS AN ENVELOPE OF CIRCLES 
EXAMPLE 42: HYPERBOLA AS AN ENVELOPE OF LINES 
EXAMPLE 43: CURVATURE OF CONIC SECTIONS
MECHANISMS 
EXAMPLE 44: A CRANK PISTON MECHANISM
EXAMPLE 45: A QUICK RETURN MECHANISM 
EXAMPLE 46: PAUCELLIER’S LINKAGE 
EXAMPLE 47: OFF CENTERED CIRCULAR CAM 
EXAMPLE 48: SINUSOIDAL MOTION FROM A RECIPROCATING ROLLER FOLLOWER 
EXAMPLE 49: GENERAL DISC CAM WITH A RECIPROCATING ROLLER FOLLOWER 
EXAMPLE 50: HARBORTH GRAPH 
SPLINE CURVES 
EXAMPLE 51: CUBIC SPLINE 
EXAMPLE 52: A TRIANGLE SPLINE 
EXAMPLE 53: ANOTHER TRIANGLE SPLINE 
CAUSTICS 
EXAMPLE 54: CAUSTICS IN A CUP OF COFFEE 
EXAMPLE 55: A NEPHROID BY ANOTHER ROUTE 
EXAMPLE 56: CAUSTIC IN AN ELLIPSE 
EXAMPLE 57: COFFEE CUP CAUSTICS REVISITED – FINITE LIGHT SOURCE 
EXAMPLE 58: TSCHIRNHAUSEN’S CUBIC 
EXAMPLE 59: GENERAL CAUSTIC (PARALLEL RAYS) 
EXAMPLE 60: GENERAL CAUSTIC (POINT LIGHT SOURCE) 
CURVES 
EXAMPLE 61: ROSACE A QUATRE BRANCHES 
EXAMPLE 62: OVAL OF CASSINI 
EXAMPLE 63: OVAL OF DESCARTES 
EXAMPLE 64: PASCAL’S LIMAÇON
EXAMPLE 65: KULP QUARTIC & THE WITCH OF AGNESI 
EXAMPLE 66: NEWTON’S STROPHOID 
EXAMPLE 67: MACLAURIN’S TRISECTRIX AND OTHER SUCH LIKE 
EXAMPLE 68: TRISECTRICE DE DELANGE 
EXAMPLE 69: “FOGLIE DEL SUARDI” 
EXAMPLE 70: A CONSTRUCTION OF DIOCLETIAN 
EXAMPLE 71: KAPPA CURVE 
EXAMPLE 72: KEPLER’S EGG 
EXAMPLE 73: CRUCIFORM CURVE 
EXAMPLE 74: PEDAL CURVE OF A PARABOLA 
EXAMPLE 75: NEGATIVE PEDAL CURVE OF A PARABOLA 
EXAMPLE 76: CONTRAPEDAL CURVE OF A PARABOLA 
EXAMPLE 77: EVOLUTE OF A PARABOLA 
EXAMPLE 78: PARALLEL CURVES TO PARABOLAS 
EXAMPLE 79: LIMIT OF THE CIRCUMCIRCLE AND EXCIRCLE 
FUNCTIONS AND PARAMETRIC CURVES 
EXAMPLE 80: INTERSECTION OF TWO TANGENTS TO A QUADRATIC 
EXAMPLE 81: TANGENT TO A CUBIC 
EXAMPLE 82: AREA UNDER A CHORD OF A PARABOLA 
EXAMPLE 83: AREA OF A TRIANGLE FORMED BY A TANGENT TO THE FUNCTION Y =1/X 
EXAMPLE 84: ORTHOCENTER OF TRIANGLE DEFINED BY 3 POINTS ON THE FUNCTION Y="1/X 

EXAMPLE 85: TRANSFORMATIONS OF FUNCTIONS 
EXAMPLE 86: OFFSET CURVE 
EXAMPLE 87: EVOLUTE OF THE LOGARITHMIC SPIRAL 
EXAMPLE 88: PEDAL CURVE WHERE THE PEDAL POINT IS ON THE EVOLUTE 
MISCELLANEOUS PROBLEMS 
EXAMPLE 89: FEYNMAN’S TRIANGLE 
EXAMPLE 90: A GENERALIZATION OF FEYNMAN’S TRIANGLE 
EXAMPLE 91: MIXTILINEAR INCIRCLES AND EXCIRCLES 
EXAMPLE 92: JOINING THE CENTERS OF CIRCLES TANGENT TO 2 SIDES OF A TRIANGLE AND CENTER LYING ON THE THIRD 
EXAMPLE 93: MAXIMIZING THE ANGLE FOR A RUGBY KICK 
EXAMPLE 94: TRIANGULATION 
EXAMPLE 95: AREA OF THE PEDAL TRIANGLE 
EXAMPLE 96: FAGNANO’S ALTITUDE BASE PROBLEM 
EXAMPLE 97: REGIOMONTANUS’ MAXIMUM PROBLEM 
EXAMPLE 98: EULER’S TETRAHEDRON PROBLEM 
EXAMPLE 99: MORLEY’S THEOREM 
EXAMPLE 100: A LADDER PROBLEM 
EXAMPLE 101: AN INSCRIBABLE AND CIRCUMSCRIBABLE PENTAGON 

101 Symbolic Geometry Examples
CALCULUSE

CALCULUS EXPLORATIONS WITH GEOMETRY EXPRESSIONS

HANNAH  IRINA LYUBLINSKAYA/ HANNAH  VALERIY RYZHIK 

INTRODUCTION
CROSS REFERENCE TABLE
1. LIMITS
1.1 THE FORMAL DEFINITION OF A LIMIT
1.2 THE SQUEEZE THEOREM
1.3 AREA OF A CIRCLE
2. DERIVATIVES
2.1 EXPLORING TANGENT LINES
2.2 MEAN VALUE THEOREM
2.3 THE DERIVATIVE OF EVEN FUNCTIONS
2.4 THE DERIVATIVE OF ODD FUNCTIONS
2.5 DIFFERENTIABILITY OF A PIECEWISE FUNCTION AT A POINT
2.6 DERIVATIVE OF AN INVERSE FUNCTION
3. APPLICATIONS OF DERIVATIVES
3.1 ENVELOPE OF A PARABOLA
3.2 LINEAR APPROXIMATION
3.3 NEWTON’S METHOD
3.4 RECTANGLE IN A SEMICIRCLE
3.5 FLOATING LOG
3.6 ART GALLERY
4. INTEGRALS AND THEIR APPLICATIONS
4.1 REPRESENTATION OF THE ANTIDERIVATIVE
4.2 THE FUNDAMENTAL THEOREM OF CALCULUS
4.3 THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
4.4 INTEGRAL OF AN INVERSE FUNCTION
4.5 THE TRAPEZOIDAL METHOD
4.6 MINIMUM AREA
5. DIFFERENTIAL EQUATIONS
5.1 ORTHOGONAL TRAJECTORY TO A CIRCLE
5.2 ORTHOGONAL TRAJECTORY TO A HYPERBOLA
6. SEQUENCES AND SERIES
6.1 INFINITE STAIRS
6.2 THE SNOWMAN PROBLEM
6.3 TRIGONOMETRIC DELIGHT
6.4 CONVERGING OR DIVERGING?
7. PARAMETRIC EQUATIONS AND POLAR COORDINATES
7.1 FOLIUM OF DESCARTES USING PARAMETRIC EQUATIONS
7.2 FOLIUM OF DESCARTES IN POLAR COORDINATES

CALCULUS
101 CONICS

101 CONIC SECTIONS EXAMPLES WITH GEOMETRY EXPRESSIONS (Philip Todd)

INTRODUCTION 
ELEMENTARY EXAMPLES 

EXAMPLE 1: TWO PINS AND A PIECE OF STRING 
EXAMPLE 2: OPTICAL PROPERTY OF FOCI 
EXAMPLE 3: ELLIPSE GIVEN SEMI-MAJOR AND SEMI-MINOR AXES 
EXAMPLE 4: HYPERBOLA AS A LOCUS OF CONSTANT DIFFERENCES 
EXAMPLE 5: OPTICAL PROPERTIES OF THE HYPERBOLA 
EXAMPLE 6: CONJUGATE HYPERBOLAS 
EXAMPLE 7: PARABOLA IN TERMS OF FOCUS AND DIRECTRIX 
EXAMPLE 8: OPTICAL PROPERTIES OF THE PARABOLA 
EXAMPLE 9: GENERAL CONIC VIA FOCUS / DIRECTRIX 
EXAMPLE 10: POLAR LINE 
EXAMPLE 11: LOCUS OF INTERSECTIONS OF TANGENTS AT THE END OF CHORDS THROUGH A FIXED POINT
PARAMETRIC LOCATION 
EXAMPLE 12: PARAMETRIC LOCATION ON AN ELLIPSE 
EXAMPLE 13: CONJUGATE DIAMETERS AND PARAMETRIC LOCATION 
EXAMPLE 14: HYPERBOLA PARAMETER 
EXAMPLE 15: AXIS INTERSECTIONS OF ELLIPSE TANGENTS 
EXAMPLE 16: AXIS INTERSECTIONS OF HYPERBOLA TANGENTS 
EXAMPLE 17: CONJUGATE DIAMETERS AND HYPERBOLA PARAMETER 
EXAMPLE 18: PARAMETRIC LOCATION ON A PARABOLA 
ELLIPSE EXAMPLES 
EXAMPLE 19: LOCUS OF INTERSECTIONS OF PERPENDICULAR TANGENTS 
EXAMPLE 20: LOCUS OF INTERSECTION OF TANGENTS AT THE ENDS OF A CHORD THROUGH THE FOCUS 
EXAMPLE 21: ANGLE BETWEEN TWO TANGENTS 
EXAMPLE 22: TANGENT DIRECTION 
EXAMPLE 23: LOCUS OF MIDPOINTS OF PARALLEL CHORDS 
EXAMPLE 24: ANGLE BETWEEN SUPPLEMENTAL CHORDS 
EXAMPLE 25: FOCAL TRIANGLE 
EXAMPLE 26: LOCUS OF INTERSECTION OF TANGENTS DEFINED BY THE FOCAL TRIANGLE 
EXAMPLE 27: TRIANGLE THROUGH CENTER AND FOCUS 
EXAMPLE 28: TRIANGLE FORMED BY TANGENT AT END OF DIAMETER, AND FOCAL CHORDS 
EXAMPLE 29: PROJECTING THE FOCUS ONTO THE TANGENT 
EXAMPLE 30: FOOT OF THE NORMAL 
EXAMPLE 31: CENTRAL NORMAL 
EXAMPLE 32: LENGTH OF LINE FROM CENTER TO TANGENT PARALLEL WITH FOCAL RADIUS 
EXAMPLE 33: LATUS RECTUM 
EXAMPLE 34: DISTANCE TO FOCUS AND TO FOCAL TANGENT 
EXAMPLE 35: LOCUS OF INTERSECTION OF TANGENT WITH PARALLEL RADIUS THROUGH THE CENTER 
EXAMPLE 36: QUADRILATERAL CIRCUMSCRIBING A CENTRAL CONIC
PARABOLA EXAMPLES 
EXAMPLE 37: FIND THE CANONICAL FORM OF A PARABOLA 
EXAMPLE 38: FINDING THE PARAMETER OF A PARABOLA 
EXAMPLE 39: RELATIONSHIP BETWEEN THE LENGTHS OF PERPENDICULAR TANGENTS 
EXAMPLE 40: NORMAL AND SUBNORMAL 
EXAMPLE 41: PARABOLA TANGENTS 
EXAMPLE 42: TANGENT NORMAL AXIS TRIANGLE 
EXAMPLE 43: TANGENT INTERSECTION 
EXAMPLE 44: LINE JOINING TANGENT INTERSECTION TO FOCUS 
EXAMPLE 45: LINE FROM CHORD’S INTERSECTION WITH DIRECTRIX 
EXAMPLE 46: CIRCUMCIRCLE OF THREE TANGENT INTERSECTIONS PASSES THROUGH THE FOCUS 
EXAMPLE 47: INTERSECTION OF THE FOCUS-TANGENT PERPENDICULAR WITH THE LINE FROM VERTEX TO POINT OF CONTACT 
EXAMPLE 48: INTERSECTION OF THE FOCUS-TANGENT PERPENDICULAR WITH THE HORIZONTAL FROM THE POINT OF CONTACT 
EXAMPLE 49: ORTHOCENTER OF THE TANGENT TRIANGLE 
EXAMPLE 50: AREA OF TANGENT TRIANGLE 
EXAMPLE 51: ARCHIMEDEAN TRIANGLES AND QUADRATURE OF A PARABOLA 
EXAMPLE 52: LOCUS OF INTERSECTIONS OF TANGENTS OF GIVEN ANGLE 
EXAMPLE 53: LOCUS OF THE FOOT OF THE FOCUS-NORMAL PERPENDICULAR
EXAMPLE 54: LOCUS OF INTERSECTION OF PERPENDICULAR NORMALS 
EXAMPLE 55: COORDINATES OF THE INTERSECTION OF TWO NORMALS 
EXAMPLE 56: A THEOREM OF STEINER’S 
EXAMPLE 57: PARAMETRIC QUADRATIC 
EXAMPLE 58: HOW TO FIND A PARABOLA’S VERTEX FROM A CHORD AND THE TANGENTS AT ITS ENDS 
HYPERBOLA EXAMPLES 
EXAMPLE 59: HYPERBOLA WITH GIVEN ASYMPTOTES 
EXAMPLE 60: PONCELET BRIANCHON HYPERBOLA PROBLEM 
EXAMPLE 61: 9 POINT CIRCLE OF A TRIANGLE INSCRIBED IN A RIGHT HYPERBOLA 
EXAMPLE 62: TANGENT INTERSECTIONS WITH ASYMPTOTES 
SOME LOCI YIELDING CONICS 
EXAMPLE 63: LOCUS OF INCIRCLES 
EXAMPLE 64: CIRCLE THROUGH A POINT WITH GIVEN INTERCEPT TO A FIXED LINE 
EXAMPLE 65: CIRCLE THROUGH A POINT WHOSE INTERCEPT WITH A GIVEN LINE SUBTENDS A FIXED ANGLE AT THAT POINT 
EXAMPLE 66: INTERSECTION OF THE PERPENDICULAR FROM THE CENTER TO A TANGENT WITH THE FOCAL RADIUS 
EXAMPLE 67: LOCUS OF INTERSECTION OF TANGENTS AT THE END OF CONJUGATE DIAMETERS
EXAMPLE 68: INTERSECTION OF NORMALS AT ENDS OF FOCAL CHORD 
EXAMPLE 69: LOCUS OF INTERSECTION OF DIAGONALS OF A TRAPEZIUM 
EXAMPLE 70: LOCUS OF CENTER OF SEGMENT OF TANGENT CUT OFF BY TWO FIXED TANGENTS 
EXAMPLE 71: NEWTON’S CONIC CONSTRUCTION 
EXAMPLE 72: MACLAURIN’S CONIC CONSTRUCTION 
SOME EXAMPLES INVOLVING ENVELOPES 
EXAMPLE 73: ENVELOPE OF HYPOTENUSE OF RIGHT ANGLED TRIANGLE, WHOSE SHORT SIDES ADD UP TO A CONSTANT 
EXAMPLE 74: ENVELOPE OF HYPOTENUSE OF RIGHT ANGLED TRIANGLES OF CONSTANT AREA
EXAMPLE 75: A TRIANGLE WHOSE VERTICES TRAVERSE FIXED LINES AND TWO OF WHOSE SIDES PASS THROUGH FIXED POINTS 
EXAMPLE 76: ENVELOPE OF HYPOTENUSE OF RIGHT ANGLED TRIANGLES WITH CONSTANT PERIMETER 
EXAMPLE 77: THE ENVELOPE OF THE LINE THE PRODUCT OF WHOSE DISTANCES FROM TWO FIXED POINTS IS CONSTANT 
EXAMPLE 78: A LINE AT FIXED ANGLE TO A SEGMENT FROM A FIXED POINT TO A FIXED LINE 
EXAMPLE 79: CROSS SECTION OF A HYPERBOLOID 
EXAMPLE 80: SIDES OF REFLECTED TRIANGLE 
CENTERS OF CURVATURE, EVOLUTES AND CAUSTICS 
EXAMPLE 81: CENTERS OF CURVATURE OF AN ELLIPSE 
EXAMPLE 82: RADIUS OF CURVATURE IN TERMS OF NORMAL AND SEMI-PARAMETER 
EXAMPLE 83: CENTERS OF CURVATURE OF A HYPERBOLA 
EXAMPLE 84: CENTERS OF CURVATURE OF A PARABOLA 
EXAMPLE 85: FOCAL CHORD OF CURVATURE A PARABOLA 
EXAMPLE 86: CONSTRUCTION FOR CENTER OF CURVATURE OF AN ELLIPSE 
EXAMPLE 87: EVOLUTE EQUATION OF A PARABOLA 
EXAMPLE 88: PARABOLA CAUSTIC WITH INCOMING LIGHT PERPENDICULAR TO THE AXIS 
EXAMPLE 89: PARABOLA CAUSTIC FROM PARALLEL RAYS SKEW TO THE AXIS 
SYNTHETIC METHODS 
EXAMPLE 90: AXIS INTERSECTIONS OF A CONIC 
EXAMPLE 91: EQUATION OF THE CONIC THROUGH FIVE GIVEN POINTS 
EXAMPLE 92: FAMILY OF CONICS WHICH INTERSECT A GIVEN CONIC IN THE SAME POINTS AS A PAIR OF LINES 
EXAMPLE 93: CONIC MAKING DOUBLE CONTACT WITH A GIVEN CONIC
EXAMPLE 94: CONIC GIVEN TWO TANGENTS AND CHORD OF CONTACT 
PAIRS OR FAMILIES OF CONICS 
EXAMPLE 95: CONFOCAL CONICS INTERSECT AT RIGHT ANGLES 
EXAMPLE 96: A LINE CUTTING TWO SIMILAR CONCENTRIC CONICS 
EXAMPLE 97: ANGLE BETWEEN TANGENTS TO AN INNER AND OUTER CONCENTRIC SIMILAR ELLIPSE 
EXAMPLE 98: LOCUS OF TANGENT POINTS TO A FAMILY OF CONFOCAL CONICS 
EXAMPLE 99: POINT OF CONTACT OF CONFOCAL CONIC TANGENT TO A FAMILY OF PARALLEL LINES 
EXAMPLE 100: POINT OF CONTACT OF CONIC TANGENT TO A FAMILY OF LINES PASSING THROUGH A POINT ON THE AXIS 
EXAMPLE 101: LOCUS OF CENTERS OF CONICS WHICH PASS THROUGH A TRIANGLE AND ITS ORTHOCENTER

101 Conic Sections Examples
ALGEBRA GEOMETRY

CONNECTING ALGEBRA AND GEOMETRY THROUGH TECHNOLOGY  JIM WIECHMANN

INTRODUCTION 
PROFESSIONAL DEVELOPMENT UNIT FOR GEOMETRY EXPRESSIONS USING

A CONSTRAINT APPROACH IN TEACHING MATHEMATICS 
INTRODUCTION 
GUIDE TO THE SCREEN 
LESSON 1: USING THE CONSTRAINT APPROACH 
LESSON 2: DRAWING GEOMETRIC FIGURES 
LESSON 3: SYMBOLIC OUTPUTS AND REAL OUTPUTS 
LESSON 4: VARIATION AND ANIMATION 
LESSON 5: FUNCTIONS AND LOCI 
APPENDIX: TROUBLESHOOTING 
APPLYING GEOMETRY EXPRESSIONS IN THE ALGEBRA 2 AND PRE-CALCULUS CLASSROOMS
UNIT 1: PARAMETRIC FUNCTIONS 
LESSON 1: A QUICK REVIEW OF FUNCTIONS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 2: DUDE, WHERE’S MY FOOTBALL? 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 3: GO SPEED RACER! 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 4: PARAMETRIC PROBLEMS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
UNIT 2: CONICS AND LOCI 
LESSON 1: INTRODUCING LOCI 
LEARNING OBJECTIVES 
STUDENT WORKSHEETS 
LESSON 2: THE CIRCLE
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 3: THE ELLIPSE 
LEARNING OBJECTIVES 
STUDENT WORKSHEETS 
LESSON 4: CONICS AND ENVELOPE CURVES 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 5: INSIDE OUT ELLIPSES 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 6: ECCENTRICITY 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 

Connecting Algebra and Geometry through Technology
GEOMETRY PROOF

DEVELOPING GEOMETRY PROOFS WITH GEOMETRY EXPRESSIONS™

IRINA LYUBLINSKAYA, VALERIY RYZHIK, DAN FUNSCH

I. INTRODUCTION
II. HOW TO USE THESE MATERIALS
CROSS REFERENCE TABLE
III. DESCRIPTION OF PROOF METHODS
GEOMETRIC METHOD
1. SEGMENTS IN A SQUARE – PURE GEOMETRY
ALGEBRAIC METHOD
2. SEGMENTS IN A SQUARE – ALGEBRA FOR ALL
COORDINATE METHOD
3. SEGMENTS IN A SQUARE – ON A COORDINATE PLANE
VECTOR METHOD
4. SEGMENTS IN A SQUARE – FOLLOW THE VECTOR
TRANSFORMATIONS METHOD
5. SEGMENTS IN A SQUARE – NEW SPIN!
IV. RELATIONSHIPS BETWEEN GEOMETRIC FIGURES
COLLINEARITY
6. DIAMETER OF A CIRCLE
7. ANGLE BISECTORS OF A TRAPEZOID
PARALLELISM
8. EXTERIOR ANGLE BISECTOR
PERPENDICULARITY
10. TWO INTERSECTING CIRCLES
11. TWO ANGLE BISECTORS
TYPES OF GEOMETRIC FIGURES
12. TWO TANGENT CIRCLES
13. IT’S A TRAPEZOID
14. SHAPE OF A QUADRILATERAL
15. QUADRILATERAL IN A TRAPEZOID
17. RECTANGLES WITH EQUAL AREAS
18. SLIDING SEGMENT
19. UNEXPECTED LOCUS
V. RELATIONSHIPS BETWEEN MEASURES OF GEOMETRIC FIGURES
SEGMENT LENGTH
20. LENGTH OF A COMMON TANGENT
21. TWO MEDIANS IN A TRIANGLE
22. INSCRIBED TRAPEZOID
23. TWO CONCENTRIC CIRCLES
24. RIGHT TRIANGLE INEQUALITY
25. DIAGONAL OF A PARALLELOGRAM
26. SEGMENT IN A TRIANGLE
27. POINT INSIDE AN EQUILATERAL TRIANGLE
28. TRIANGLE WITH THE SMALLEST PERIMETER
29. POWER OF A POINT THEOREM
30. SHORTEST PATH
31. TWO EQUAL CIRCLES
AREA
32. AREA OF A QUADRILATERAL
33. QUADRILATERAL IN A SQUARE
34. AREA COMPARISON
35. TRIANGLE IN A SQUARE
36. TRIANGLE WITH THE LARGEST AREA
ANGLE
37. SEGMENTS IN AN EQUILATERAL TRIANGLE
38. STAR259
39. ANGLE COMPARISON
VI. RECONSTRUCTION PROBLEMS
40. TRIANGLE FROM THREE MIDPOINTS
41. RESTORING AN EQUILATERAL TRIANGLE
42. TRIANGLE FROM THREE MEDIANS 
43. FROM TRAPEZOID TO SQUARE
VII. CONCLUSION

Developing Geometry Proofs with Geometry Expressions
HIGH SCHOOL

EXPLORING WITH GEOMETRY EXPRESSIONS IN HIGH SCHOOL MATHEMATICS IAN SHEPPARD

CHAPTER 1 – INTRODUCTION
CHAPTER 2 – CONSTRAINTS
CHAPTER 3 – ALGEBRA IN GX (SYMBOLICS)
CHAPTER 4 – DANNY’S ROOM

CHAPTER 5 – CONGRUENCE
LAB # 1 THREE SIDES
LAB # 2 TWO SIDES AND AN ANGLE
LAB # 3 HYPOTENUSE AND A SIDE
LAB # 4 TWO ANGLES AND A SIDE
LAB # 5 THREE ANGLES
CHAPTER 6 – PROOF
LAB # 6 CONGRUENT TRIANGLES - SSS
LAB # 7 BISECT AN ANGLE
LAB # 8 CONSTRUCT A PARALLEL LINE 46
LAB # 9 COPY AN ANGLE
LAB # 10 PERPENDICULAR BISECTOR
LAB # 11 CENTRAL ANGLE THEOREM
LAB # 12 OTHER CIRCLE THEOREMS
LAB # 13 THE “CENTER” OF A TRIANGLE
CHAPTER 7 – SLIDE, TURN, FLIP, AND RESIZE
LAB # 14 SLIDE
LAB # 15 CLIMBING UP!
LAB # 16 SLIDING WITH COORDINATES
LAB # 17 TURN RIGHT
LAB # 18 TURNING WITH COORDINATES
LAB # 19 FERRIS WHEEL
LAB # 20 FLIPPING OVER
LAB # 21 MULTIPLE FLIPS
LAB # 22 KALEIDOSCOPE
LAB # 23 FLIPPING AND COORDINATES
LAB # 24 RESIZE
LAB # 25 RESIZE AND COORDINATES
LAB # 26 DETERMINING THE DILATION
CHAPTER 8 – RIGHT INTO TRIANGLES
LAB # 27 PYTHAGOREAN THEOREM
LAB # 28 CONVERSE OF THE PYTHAGOREAN THEOREM
LAB # 29 RIGHT TRIANGLES - 30° AND 45°
LAB # 30 MEASURES OF “STANDARD” TRIANGLES
LAB # 31 USE OF SIMILAR TRIANGLES TO SOLVE RIGHT TRIANGLES
LAB # 32 TARGET PRACTICE
LAB # 33 SINE FORMULA
LAB # 34 LAW OF COSINES
LAB # 35 SOLVING NON-RIGHT TRIANGLES
LAB # 36 HOW MUCH HIGHER?
LAB # 37 STRANDED ON AN ISLAND

CHAPTER 9 – COORDINATE GEOMETRY
LAB # 38 DISTANCE FORMULA
LAB # 39 MIDPOINT FORMULA
LAB # 40 MIDPOINTS OF A QUADRILATERAL
LAB # 41 DIAGONALS OF A RHOMBUS
LAB # 42 ANGLE IN A SEMICIRCLE
LAB # 43 THE LAW OF COSINES
LAB # 44 SUBTRACTION FORMULA
Exploring with Geometry Expressions

CHAPTER 10 – LOCI
LAB # 45 THE CIRCLE
LAB # 46 SHAKE IT MAMA
LAB # 47 DISTANCE FROM TWO POINTS
LAB # 48 DETECTIVE WORK?
LAB # 49 TURKEY TETHER
LAB # 50 PARABOLIC FOCUS
LAB # 51 BEZIER CURVES

REFERENCES
APPENDIX A – GX AND FUNCTIONS
APPENDIX B - INSIGHT WITH GEOMETRY EXPRESSIONS
INTRODUCTION
WARM UP
A SEQUENCE OF ALTITUDES
ANGLES AND CIRCLES
TRIANGULATION
RECTANGLE CIRCUMSCRIBING AN EQUILATERAL TRIANGL
AREA OF A HEXAGON BOUNDED BY TRIANGLE SIDE TRISECTORS
AN INVESTIGATION OF INCIRCLES, CIRCUMCIRCLES AND RELATED MATTERS
CIRCUMCIRCLE RADIUS
INCIRCLE RADIUS
INCIRCLE CENTER IN BARYCENTRIC COORDINATES
HOW DOES THE POINT OF CONTACT WITH THE INCIRCLE SPLIT A LINE?
EXCIRCLES

COURSE II
FUNC TRANS
SECONDARY SCHOOL
AGRI MATHS I
AGRI MATHS II
TORTOISE
ATLAS LINKAGES
COURSE II
 COURSE BY MATHEMATICAL TECHNOLOGY ON APPLICATION
FUNCTION TRANSFORMATIONS TIM BROWN
USING SYMBOLIC GEOMETRY TO TEACH SECONDARY SCHOOL MATHEMATICS - GEOMETRY EXPRESSIONS ACTIVITIES FOR ALGEBRA 2 AND PRECALCULUS IRINA LYUBLINSKAYA, VALERIY RYZHIK
THE FARMER AND THE MATHEMATICIAN: USING GEOMETRY EXPRESSIONS™ AND GOOGLE™ EARTH TO INVESTIGATE CROP CIRCLES - I LARRY OTTMAN
 THE FARMER AND THE MATHEMATICIAN: USING GEOMETRY EXPRESSIONS™ AND GOOGLE™ EARTH TO INVESTIGATE CROP CIRCLES - II  LARRY OTTMAN

THE TORTOISE AND ACHILLES USING GEOMETRY EXPRESSIONS™ TO INVESTIGATE THE INFINITE LARRY OTTMAN

ATLAS OF THE FOUR-BAR LINKAGE EUGENE FICHTER, PHILIP TODD, DIETER MUELLER 

FUNC TRANS

FUNCTION TRANSFORMATIONS TIM BROWN 

INTRODUCTION 
UNIT 1: INTRODUCTION TO TRIGONOMETRY

LESSON 1: RIGHT TRIANGLE TRIGONOMETRY 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 2: THE UNIT CIRCLE 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
UNIT 2: FUNCTION TRANSFORMATIONS 
LESSON 1: VERTICAL TRANSLATIONS OF FUNCTIONS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 2: VERTICAL DILATIONS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 3: COMBINED VERTICAL TRANSFORMATIONS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 4: CIRCULAR AND HARMONIC MOTION 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 5: HORIZONTAL AND COMBINED TRANSFORMATIONS 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
LESSON 6: SINUSOIDAL CURVES 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
EXTENSION A: CIRCLES AND ELLIPSES 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
EXTENSION B: ABSOLUTE VALUE 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
EXTENSION C: COSINE AND TANGENT 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 
EXTENSION D: VERTICAL ASYMPTOTES 
LEARNING OBJECTIVES 
OVERVIEW FOR THE TEACHER 
STUDENT WORKSHEETS 

Function Transformations
SECONDARY SCHOOL

USING SYMBOLIC GEOMETRY TO TEACH SECONDARY SCHOOL MATHEMATICS - GEOMETRY EXPRESSIONS ACTIVITIES FOR ALGEBRA 2 AND PRECALCULUS IRINA LYUBLINSKAYA, VALERIY RYZHIK

INTRODUCTION 
DISCOVERING PARABOLAS 

PART 1 – PARABOLA BY 3 POINTS
PART 2 – THE EXISTENCE OF A PARABOLA PASSING THROUGH THREE ARBITRARY POINTS 
EXTENSIONS
SOLVING SYSTEMS OF EQUATIONS (INEQUALITIES) WITH PARAMETERS 
PART 1 – SETTING UP THE PROBLEM IN GEOMETRY EXPRESSIONS 
PART 2 – WHEN THE SYSTEM HAS NO SOLUTIONS 
PART 3 – INVESTIGATION OF THE NUMBER OF SOLUTIONS
PART 4 – SOLVING SYSTEMS OF EQUATIONS 
PART 5 – SOLVING SYSTEMS OF INEQUALITIES WITH PARAMETERS
EXTENSIONS: 
STAINED GLASS DESIGN 
PART 1 – SETTING UP PROBLEM IN GEOMETRY EXPRESSIONS 
PART 2 – CREATING STAINED GLASS DESIGN
PART 3 – FINDING EQUATIONS OF THE CURVES IN THE STAINED GLASS DESIGN
PART 4 – VERIFICATION OF THE EQUATIONS WITH GEOMETRY EXPRESSIONS
EXTENSIONS 
TRANSLATION ALONG COORDINATE AXES 
PART 1 – TRANSLATION ALONG THE Y-AXIS 
PART 2 – TRANSLATION ALONG THE X-AXIS 
PART 3 – COMMUTATIVE PROPERTY OF TRANSLATION 
PART 4 – APPLICATIONS AND ASSESSMENT PROBLEMS 
EXTENSION:
A LITTLE TRIG
PART 1 – INVESTIGATING AREA OF THE TRIANGLE
PART 2 – OPTIMIZING THE PERIMETER OF THE RECTANGLE
ONE HYPERBOLA 
PART 1 – INVESTIGATING AREA OF THE RECTANGLE
PART 2 – OPTIMIZING PERIMETER OF THE RECTANGLE
PART 3 – OPTIMIZATION OF THE DIAGONAL OF THE RECTANGLE
THREE EXTREMA (CIRCLE) 
PART 1 – LENGTH OF A TANGENT SEGMENT TO A CIRCLE
PART 2 – AREA OF A TRIANGLE FORMED BY A TANGENT LINE AND THE COORDINATE AXES
PART 3 – PERIMETER OF A RECTANGLE WHOSE DIAGONAL IS A TANGENT SEGMENT
TWO PARABOLAS 
PART 1 – OPTIMIZING PERIMETER OF RECTANGLE
PART 2 – OPTIMIZING THE DIAGONAL OF THE RECTANGLE 
PART 3 – COMPARISON OF POINTS OF EXTREMA FOR PERIMETER, DIAGONAL, AND AREA

Using Symbolic Geometry
AGRI MATHS I

THE FARMER AND THE MATHEMATICIAN: USING GEOMETRY EXPRESSIONS™ AND GOOGLE™ EARTH TO INVESTIGATE CROP CIRCLES LARRY OTTMAN

LESSON ONE 
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
LESSON TWO
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
LESSON THREE: THE ALGEBRA FARMER
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
LESSON FOUR
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
EXCITING EXTENSIONS 
DRAMATIC DIVERSION
LESSON FIVE
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
EXCITING EXTENSION
LESSON SIX: THE ALGEBRA FARMER II 
INFORMATIVE INSTRUCTIONS
LESSON SEVEN
QUICK QUESTIONS
INFORMATIVE INSTRUCTIONS
TABLE OF CONTENTS
REPORTING RESULTS 
EXCITING EXTENSION
LESSON EIGHT
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
EXCITING EXTENSION
EXHILARATING EXTENSION 
TAXING TRIGONOMETRIC TREK
LESSON NINE: THE GREEDY FARMER 
INFORMATIVE INSTRUCTIONS
REPORTING RESULTS 
TEACHER NOTES
INTRODUCTION
OBJECTIVES 

LESSON ONE
LESSON NOTES 
OBJECTIVES 
LESSON TWO
LESSON NOTES 
LESSON THREE
LESSON NOTES 
OBJECTIVES 
LESSON FOUR
LESSON NOTES 
OBJECTIVES 
LESSON FIVE 
LESSON NOTES 
LESSON SIX
OBJECTIVES 
LESSON NOTES 
OBJECTIVES 
LESSON SEVEN 
LESSON NOTES 
LESSON EIGHT 
OBJECTIVES 
LESSON NOTES 
LESSON NINE
OBJECTIVES 
LESSON NOTES 
FURTHER INVESTIGATION

The Farmer and the Mathematician
AGRI MATHS II

THE FARMER AND THE MATHEMATICIAN: USING GEOMETRY EXPRESSIONS™ AND GOOGLE™ EARTH TO INVESTIGATE CROP CIRCLES II  LARRY OTTMAN

The Farmer is back and he’s been collecting a lot of questions for his new mathematician. (The mathematician from our last book is “on vacation - practicing his zero slope!”) Our farmer is still using his Center Pivot Irrigation System – those long stretches of wheeled scaffolding, sometimes up to a half mile in length, rotating around a fixed point attached to his water source. And he’s become quite good at scanning Google Earth to see what his fellow farmers are up to.

Again, as in the first book, The Farmer and the Mathematician, we demonstrate to the student that Mathematics is not just for calculating the price of produce in
the grocery store. Here we have a fresh batch of examples of mathematics in action using our symbolic geometry software, Geometry Expressions, with
images from Google Earth.
The process of mathematical modeling is both critical to using mathematics in the world around us, and to developing an understanding and appreciation for the
true utility of mathematics. A mathematician or engineer would first be given a problem to solve. They would also be given constraints, or a set of rules
and conditions that must be followed. From that, the scientist would attempt to create a mathematical representation of the problem. Often, real problems are
much too complex and it is necessary to constrain, or simplify them further. This is very similar to the approach that a student would take to gain insight into a
problem by solving a simpler, related one. Geometry Expressions has been written to mirror this process. It is a constraint-based geometric and algebraic
modeling program that allows the student to investigate problems in both a numeric and symbolic representation.
So join the Farmer and his mathematical instructor, Sophie, as they fly over Texas, investigating interesting geometrical ideas and stumbling over important
mathematical concepts. The content is appropriate and adaptable for students in a range of courses from Algebra and Geometry up through and including
Calculus.

The Farmer and the Mathematician II
TORTOISE

THE TORTOISE AND ACHILLES USING GEOMETRY EXPRESSIONS™ TO INVESTIGATE THE INFINITE LARRY OTTMAN

LESSON ONE
LESSON TWO
LESSON THREE
LESSON FOUR
LESSON FIVE
LESSON SIX
LESSON SEVEN
LESSON EIGHT
LESSON NINE
LESSON TEN
TEACHER NOTES
LESSON ONE
LESSON
TWO LESSON
THREE LESSON
FOUR LESSON 

FIVE LESSON

SIX LESSON

SEVEN LESSON

EIGHT LESSON

NINE LESSON

TEN LESSON

The Tortoise and Achilles
ATLAS LINKAGES

ATLAS OF THE FOUR-BAR LINKAGE EUGENE FICHTER, PHILIP TODD, DIETER MUELLER 

1. INTRODUCTION
THERE ARE MANY SITUATIONS IN MACHINE DESIGN WHICH REQUIRE PARTS TO MOVE ALONG COMPLEX PATHS. CAMS AND LINKAGES ARE TWO COMMONLY USED DEVICES FOR PRODUCING SUCH MOVEMENT. EACH OF THESE DEVICES HAS ADVANTAGES AND DISADVANTAGES, BUT FOR MANY DESIGNS THE ADVANTAGES OF A LINKAGE ARE SUBSTANTIAL IF THE DIFFICULTY OF DESIGNING THE LINKAGE CAN BE OVERCOME. THE OBJECTIVE OF THIS ATLAS IS TO SIMPLIFY THE TASK OF DESIGNING FOUR-BAR LINKAGES WITH EITHER PIN JOINTS OR SLIDERS 
2. CRANK ROCKER

3. CRANK CRANK

4. CRANK SLIDER

5. INVERTED CRANK SLIDER 

Electronic Atlas of the Four Bar Linkage
DRAW QUICKLY
ENGINEERING
CAM AND FOLLOWER
PAUCELLIER
GEOMETRIC MODELING
Electronic Atlas of the Four Bar Linkage
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