Circular Geometry Robust Constructions Proofs Chapter 4 Axiom Systems: Ancient and Modern Approaches • Euclid’s definitions A point is that which has no part A.
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Transcript Circular Geometry Robust Constructions Proofs Chapter 4 Axiom Systems: Ancient and Modern Approaches • Euclid’s definitions A point is that which has no part A.
Circular Geometry
Robust Constructions
Proofs
Chapter 4
Axiom Systems:
Ancient and Modern Approaches
• Euclid’s definitions
A point is that which has no part
A line is breadthless length
A straight line is a line which lies evenly with
the points on itself
etc.
• We need clarification
Axiom Systems:
Ancient and Modern Approaches
• David Hilbert redefined, clarified
Cleaned up ambiguities
• Basic objects of geometry
point
line
considered undefined terms
plane
• Many geometry texts use Hilbert’s axioms
Language of Circles
• Definition:
Set of points
Fixed distance from point A
Distance called the radius
A called the center
• Interior: P : d ( A, B) r
• Exterior: P : d ( A, B) r
Language of Circles
• Chord of a circle:
Line segment joining
two points on the circle
• Diameter: a chord
containing the center
• Tangent: a line containing exactly one
point of the circle
Will be perpendicular to radius at that point
Language of Circles
• Circumference:
Length of the
perimeter
• Sector:
Pie shaped portion
bounded by arc and
two radii
Language of Circles
• Segment:
Region bounded by
arc and chord
• Central angle
CAB, center is the
angle vertex
Language of Circles
• Inscribed angle
CDB Vertex is on the
circle
Also called an angle
subtended by chord CB
Inscribed Angles
• Recall results of recent activity 1.4, 1.5 …
• Note fixed relationship between central
and inscribed angle subtending same arc
Language of Circles
• What would the conjecture of Activity 1.7
have to do with this figure from Activity 1.9
• What conjecture would you make here?
Language of Circles
• Recall conjecture made in Activity 1.8
• This also is a consequence of what we
saw in Activity 1.7
Language of Circles
• Any triangle will be cyclic (vertices lie on a
circle)
• Is this true for any four non collinear
points?
Language of Circles
• Some quadrilaterals will be cyclic
• Again, note the properties of such a
quadrilateral
Language of Circles
• Using the results of this activity
• Construct a line through a point exterior to
a circle and tangent to a circle
Robust Constructions
Developing visual proof
• Distinction between
“drawing” and
“construction”
• In Sketchpad and Goegebra
Allowable constructions based on Euclid’s
postulates
• Constructions develop visual proof
Guide us in making step by step proofs
Step-by-Step Proofs
• Each line of the proof presents
A new idea or concept
• Together with previous steps
Produces new result
Allowable Argument Justifications
• Site the given conditions
• Base argument on
Definitions
Postulates and axioms
• Constructions implicitly linked to axioms,
postulates
Allowable Argument Justifications
• Any previously proved theorem
• Previous step in current proof
• “Common notions”
Properties of equality, congruence
Arithmetic, algebraic computations
Rules of logic
Methods of Proof
1. Start by being sure of what is given
2. Clearly state the conjecture or theorem
a) P Q
b)
If hypotheses then conclusion
3. Note the steps of Geogebra construction
a) Steps of proof may well follow similar order
4. Proof should stand up to questioning of
colleagues
Direct Proof
• Start with given, work step by step towards
conclusion
• Goal is to show P Q using modus
ponens
Based on P, show sufficient conditions to
conclude Q
• Use syllogism: P R, R S, S Q
then
PQ
Indirect Proof
• Use logic role of modus tollens
P Q is equivalent to Q P
• We assume Q
• Then work step by step to show that P
cannot be true
That is P
Indirect Proof
• Alternatively we use this fact
P Q P Q
• Begin by assuming P and not Q
• Use logical reasoning to look for
contradiction
• This gives us P Q
• Which means that P Q must be true
Counter Examples
• Consider a conjecture you make P Q
You create a Geogebra diagram to illustrate
your conjecture
• Then you discover a specific example
where all the requirements of P hold true
But Q is definitely not true
• This is a counter example to show that
P Q
If-And-Only-If Proofs
• This means that P Q and
• Also written P Q
QP
• Proof must proceed in both directions
Assume P, show Q is true
Assume Q, show P is true
Proofs
• Constructed diagrams provide
visual proof
demonstration of geometric theorems
• Consider this diagram
• How might it help
us prove that the
non adjacent angles of
a cyclic quadrilateral
are supplementary.
Proof of Theorem 4.3
• Assume ABCD cyclic
• Consider pair of non-adjacent
angles BAD, BCD
• Let arc BAD be arc subtending
angle a, BCD be arc subtending b
1
• We know a + b = 360 and 2 (a b) 180
• Also ½ a = BAD , ½ b = BCD
• So BAD BCD 180
• And they are supplementary
Incircles and Excircles
• Consider concurrency of angle bisectors of
exterior angles
• Perpendicular
PJ gives radius
for excircle
• Note the other
exterior angles
are congruent
• How to show tangency points M and N?
Incircles and Excircles
• Proof : Drop perpendiculars from P to
lines XY and
XZ
• Look for
congruent
right triangles
• Finish the
proof
Families of Circles
• Orthogonal circles:
Tangents are
perpendicular at
points of intersection
• Describe how you
constructed these in Activity 8.
• How would you construct more circles
orthogonal to circle A?
Orthogonal Circles
• Describe what happens when point Q
approaches infinity.
Families of Circles
• Circles that share a common chord
• Note centers are collinear
Use this to construct more circles with chord AB
The Arbelos and Salinon
• Figures bounded by semicircular arcs
• What did you discover about the arbelos in
Activity 7?
The Arbelos and Salinon
• Note the two areas – the arbelos and the
circle with diameter RP
The Arbelos
Algebraic proof
• Calculate areas of all
the semicircles
• Calculate the are of circle
with diameter RP
• Show equality
Power of a Point
• We are familiar with the concept of a
2
3
function
f ( x) 6 x 9 x
Examples:
g ( x, y) 3x 7 xy 4 y 2
• Actually Geogebra commands are
functions that take points and/or lines as
parameters
Power of a Point
• Consider a mathematical function
involving distances with a point and a
circle
Power of a Point
• This calculation clams to be another way
to calculate Power (P, C)
Power of a Point
• Requirements for calculating Power (P, C)
Given radius, r, of circle O and distance d
(length of PO) use
d r
2
or …
2
Given line intersecting circle O at Q and R
with collinear point P use
PQ PR
Point at Infinity
• Recall stipulation P
cannot be at O
• As P approaches
O, P’ gets infinitely
far away
• This is a “point at infinity” (denoted by )
• Thus Inversion(O, C) =
The Radical Axis
• Consider Power as a measure of
Distance d from P to given circle
• If radius = 0, Power is d2
• Consider two circles, centered at A, B
Point P has a power for each
There will be some points P where Power is
equal for both circles
Set of such points called radical axis
The Radical Axis
• Set of points P with Power equal will be
bisector of segment AB
• Construction when circles do not
intersect?
The Radical Axis
• Suppose three circles are given for which
the centers are not collinear.
Each pair of circles determines a radical axis,
These three radical axes are concurrent.
Nine-Point Circle (2nd Pass)
• Recall circle which intersects feet of
altitudes (Activity 2.8)
Nine-Point Circle
• Note all the points which lie on this circle
Nine-Point Circle
• Additional phenomena
Nine point circle tangent to incircle and
excircles
Circular Geometry
Robust Constructions
Proofs
Chapter 4