Solve First – ask questions later -...
Transcript of Solve First – ask questions later -...
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Solve First – ask questions laterPhilip Todd,
Saltire Software
Please turn your smart phones ON…and browse to this location:
(If, like me, your phone sits on your desk at home, don’t worry.. These are supplemental interactive materials and not essential to the presentation.)
Browse to …
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Solve First – ask questions laterPhilip Todd,
Saltire Software
Please turn your smart phones ON…and browse to this location:
(If, like me, your phone sits on your desk at home, don’t worry.. These are supplemental interactive materials and not essential to the presentation.)
Browse to …
http://goo.gl/D14OQL
http://www.geometryexpressions.com/images/logo.gifhttp://www.geometryexpressions.com/images/logo.gif
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Problem Solving Paradigms
Problem SolutionModelBy hand computation
Mathematical insight
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Problem Solving Paradigms
Problem SolutionModelBy hand computation
Problem Solution
Brute force computation
Mathematical insight
Mathematical insight
Model
Exploration of solution space
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MuPad
Derive
TI nSpire
Maxima
ClassPad Manager
… (MathML)http://goo.gl/D14OQL
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South Bohemia Mathematical Letters Volume 22, (2014), No. 1, 26–37.
THE SIMSON-WALLACE LOCUS IN PLANE AND SPACE
PAVEL PECH1, EMIL SKRˇ´ISOVSKˇ Y´2
Abstract. In the paper several theorems related to the well-known Simson– Wallace theorem are given. Some properties of the nine-point circle and circumcircle of a given triangle are investigated. Further the relation between two Simson lines is studied, obtaining Half Angle Theorem. Special attention is paid to Steiner Deltoid curve as the envelope of the system of Simson-Wallace lines whose equation was derived. Simultaneously the generalization of the theorem into space is described and further examined.
1. Introduction
The Simson–Wallace theorem describes an interesting property regarding the points on the circumcircle of the triangle [1, 3, 4, 9].
Theorem 1.1 (Simson-Wallace). Let ABC be a triangle and P a point in plane. The feet of the perpendicular lines onto the sides of the triangle are collinear if and only if P lies on the circumcircle of ABC, Fig. 1.
Figure 1. The Simson-Wallace Theorem – points T,U,V are collinear.
Proof. To prove the theorem, we use analytical methods. We adopt a coordinate system where A = [0,0], B = [b,0] and C = [c1,c2], Fig. 2. We denote the three
Received by the editors . 1991 Mathematics Subject Classification. Key words and phrases. Simson-Walace theorem, family of Simson lines, Steiner deltoid, Cayley’s cubic.
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Theorem 1.1 (Simson-Wallace). Let ABC be a triangle and P a point in plane. The feet of the perpendicular lines onto the sides of the triangle are collinear if and only if P lies on the circumcircle of ABC, Fig. 1.
Figure 1. The Simson-Wallace Theorem – points T,U,V are collinear.
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Theorem 1.2 (Gergonne). Let ABC be a triangle and P a point in plane. If P lies on a circle concentric to the circumcircle, then the area of the pedal triangle TUV is constant, Fig. 3.
Corollary 1.4. The pedal triangle degenerates into the line if and only if P lies on the circumcircle.
Figure 3. Gergonne’s generalization – area of TUV is constant.
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Project Characteristics
Problems I did not know the answer to.Relevance inside mathematics.Relevance outside of mathematics.Discover new mathematics.
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Discovering a New World?
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Discovering a New World?
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Discovering New Mathematics?
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Discovering New Mathematics?
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Keep your eyes open for…
3 succinct geometry problems
1 nice calculus problem
differential geometry sneaking in while we aren’t looking
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary reflectors
2011: Chaotic dynamic systems
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What happens to the circumcircle of a triangle when the three points coalesce?
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X=f(T)
Y=g(T)
A
B
C
D
t
-h+t
h+t
radius Þ (-f(t)+f(h+t))2+(-g(t)+g(h+t))2 · (-f(t)+f(-h+t))2+(-g(t)+g(-h+t))2 · (-f(h+t)+f(-h+t))2+(-g(h+t)+g(-h+t))2
2·(-f(h+t)·g(t)+f(-h+t)·g(t)+f(t)·g(h+t)-f(-h+t)·g(h+t)-f(t)·g(-h+t)+f(h+t)·g(-h+t))
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X=f(T)
Y=g(T)
A
B
Ct
-h+t
h+t
D
E
F
G
r9Þ
-f(t)2
+f(h+t)
2
2
+-g(t)
2+
g(h+t)2
2
·-f(t)2
+f(-h+t)
2
2
+-g(t)
2+
g(-h+t)2
2
·-f(h+t)
2+
f(-h+t)2
2
+-g(h+t)
2+
g(-h+t)2
2
-f(h+t)·g(t)2
+f(-h+t)·g(t)
2+
f(t)·g(h+t)2
-f(-h+t)·g(h+t)
2-
f(t)·g(-h+t)2
+f(h+t)·g(-h+t)
2
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Anticomplentary Circle:
( ) RR 22lim = Apollonius Circle:
The radius of the Circle of Apollonius is not directly related to the circumradius.
Bevan Circle:
( ) RR 22lim = Brocard Circle:
2/)cos(2
)(sin41lim
2
RR
=
−ω
ω
Conway Circle:
( ) 0lim 22 =+ sr Cosine Circle:
( ) 0)tan(lim =ωR De Longchamps Circle:
( ) RCBAR 4coscoscos4lim =− Excircles:
( ) RCBAR 4)2/cos()2/cos()2/sin(4lim = Note that this equation only applies to the excircle opposite angle A. The excircles opposite angles B and C have radii of 0 because sin(0)=0
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->0 radius 20
-> infinite radius 2
-> constant multiple of radius of curvature
22
-> some other radius 6
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary reflectors
2011: Chaotic dynamic systems
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Q: Use your calculation for angle of entry in the model to find the focal surface
A: In figure 6, a trace on the reflected beam shows 5 optical focal points where incoming light is angled at approximately ±9, ±4.5 and 0 degrees. Students may find that the mirror produces a focal surface that is curved
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Q1: How do you define “optical focus point”
Q2: What is that curved surface?
Q3 Can we define “aberration”geometrically
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The parabolic mirror has no aberration at the center and gradually increasing aberration towards the edge.
Can we change the geometry to spread the aberration more evenly?
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X=f(T)
Y=g(T)
AC
th+t
D
E
F
z0 Þf(t)2-2·f(t)·f(h+t)+f(h+t)2+g(t)2-2·g(t)·g(h+t)+g(h+t)2 · f'(t)2+g'(t)2 · f'(h+t)2+g'(h+t)2
4·(f'(t)·f'(h+t)+g'(t)·g'(h+t))·(-f'(h+t)·g'(t)+f'(t)·g'(h+t))
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary reflectors
2011: Chaotic dynamic systems
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1. What angle should the lid be opened to?
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Solar Concentration Ratio
D
d
Solar Concentration Ratio = Dd
2
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2. What is the maximum solar concentration achievable with this box?
What angle would the lid be at when the maximum is achieved?
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Box Solar Cooker 2
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θ
AB
C Þd· 2-2·cos(θ) ·sin(θ)
| 2·sin(θ)-sin(2·θ)
1-cos(θ)>0
d
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> sin(theta)*(2-2*cos(theta))^(1/2)*d;
> diff(%,theta);
> solve(%=0,theta);
>
( )sin θ − 2 2 ( )cos θ d
+ ( )cos θ − 2 2 ( )cos θ d ( )sin θ2 d
− 2 2 ( )cos θ
,− + ( )arctan 2 2 π − ( )arctan 2 2 π
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π-arctan 2· 2
AB
C Þ8· 3·d
9
d
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AB
C
1
x
Þx2· 4-x2
2
| 4-x2 >0
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A little calculus
642
22
44
xxdxxd
−=
−⋅=
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A little calculus
642
22
44
xxdxxd
−=
−⋅=
differentiate
( )2353
382616
xxxx−=
−
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AB
C
1
2· 2· 33
Þ8· 3
9
Þ π-arccos13
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Tetrahedral Angle
31cos 1−=CEF
31cos 1−−= πDEC
5.109≈
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Focal Property of the parabola
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Converse?
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Converse
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What shape should you use for a solar cooker if you cannot reposition it to point at the sun?
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F number
f
D
F number = fD
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High F number
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Medium F number
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Low F number
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3. What F number needs repositioned least often?
(a) small
(b) medium(c) large
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arctan 16 r f + + − d4 32 f2 d2 256 f4 256 r2 f2
> simplify(diff(%,f)); 16 ( ) − d2 16 f2 r
( ) + d2 16 f2 + + − d4 32 f2 d2 256 f4 256 r2 f2
> solve(%=0,f);
,d4 −d4
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And the winner is…
(b) medium
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary reflectors
2011: Chaotic dynamic systems
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How would a catenary compare with a parabola as a solar cooker shape?
Where would the focus be?
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2011: Chaotic dynamic systems
2012: Interactive Euclid’s Elements Books 1-4 iBook
2013: Interactive Euclid’s Elements Books 1-6 iBook
2014: Interactive Atlas of the 4 Bar Linkage iBook
2015: Printamotion (www.printamotion.com)
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94. Regiomontanus’Maximum Problem
At what point on the earth’s surface does a perpendicularly suspended rod appear longest?
Johann Müller Regiomontanus1436-1476
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Isaac Newton1642-1727
Johann Müller Regiomontanus1436-1476
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Student work
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Browse to: Explore>Student Projects
and Explore>Journal of Symbolic Geometry
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Workshopfrom inductive reasoning to mathematical induction
PC laboratory 16:00
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Solve First – ask questions later�Snímek číslo 2Snímek číslo 3Snímek číslo 4Solve First – ask questions later�Problem Solving ParadigmsProblem Solving ParadigmsSnímek číslo 8Snímek číslo 9Snímek číslo 10Snímek číslo 11Snímek číslo 12Snímek číslo 13Snímek číslo 14Snímek číslo 15Snímek číslo 16Snímek číslo 17Snímek číslo 18Snímek číslo 19Snímek číslo 20Snímek číslo 21Snímek číslo 22Snímek číslo 23Snímek číslo 24Snímek číslo 25Snímek číslo 26Snímek číslo 27Snímek číslo 28Snímek číslo 29Snímek číslo 30Snímek číslo 31Snímek číslo 32Snímek číslo 33Snímek číslo 34Snímek číslo 35Snímek číslo 36Snímek číslo 37Snímek číslo 38Discovering a New World?Discovering a New World?Discovering New Mathematics?Discovering New Mathematics?Keep your eyes open for…Snímek číslo 44Snímek číslo 45Snímek číslo 46Snímek číslo 47Snímek číslo 48Snímek číslo 49Snímek číslo 50Snímek číslo 51Snímek číslo 52Snímek číslo 53Snímek číslo 54Snímek číslo 55Snímek číslo 56Snímek číslo 57Snímek číslo 58Snímek číslo 59Snímek číslo 60Snímek číslo 61Snímek číslo 62Snímek číslo 63Snímek číslo 64Snímek číslo 65Snímek číslo 66Snímek číslo 67Snímek číslo 68Snímek číslo 69Snímek číslo 70Snímek číslo 71Snímek číslo 72Snímek číslo 73Snímek číslo 74Snímek číslo 75Snímek číslo 76Snímek číslo 77Snímek číslo 78Snímek číslo 79Snímek číslo 80Snímek číslo 81Snímek číslo 82Snímek číslo 83Snímek číslo 84Snímek číslo 85Snímek číslo 86Snímek číslo 87Snímek číslo 88Snímek číslo 89Snímek číslo 90Snímek číslo 91Snímek číslo 92Snímek číslo 93Snímek číslo 94Snímek číslo 95Snímek číslo 96Snímek číslo 97Snímek číslo 98Snímek číslo 99Snímek číslo 100Snímek číslo 101Solar Concentration RatioSnímek číslo 103Snímek číslo 104Snímek číslo 105Snímek číslo 106Snímek číslo 107Snímek číslo 108A little calculusA little calculusSnímek číslo 111Tetrahedral AngleSnímek číslo 113Snímek číslo 114Snímek číslo 115Snímek číslo 116Focal Property of the parabola�Converse?�Converse�Snímek číslo 120F number�High F number�Medium F number�Low F number�Snímek číslo 125Snímek číslo 126Snímek číslo 127Snímek číslo 128Snímek číslo 129Snímek číslo 130Snímek číslo 131Snímek číslo 132Snímek číslo 133Snímek číslo 134Snímek číslo 135Snímek číslo 136Snímek číslo 137Snímek číslo 138Snímek číslo 139Snímek číslo 140Snímek číslo 141Snímek číslo 142Snímek číslo 143Snímek číslo 144Snímek číslo 145Snímek číslo 146Snímek číslo 14794. Regiomontanus’ Maximum Problem Snímek číslo 149Snímek číslo 150Snímek číslo 151Snímek číslo 152Snímek číslo 153Snímek číslo 154Snímek číslo 155Snímek číslo 156Snímek číslo 157Snímek číslo 158Snímek číslo 159Snímek číslo 160Snímek číslo 161Snímek číslo 162Snímek číslo 163Snímek číslo 164Student workSnímek číslo 166Snímek číslo 167Snímek číslo 168Snímek číslo 169Snímek číslo 170Snímek číslo 171Snímek číslo 172Snímek číslo 173