Circular GeometryRobust 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:

• Exterior:

: ( , )P d A B r

: ( , )P d A B r

Language of Circles

• Chord of a circle:Line segment joiningtwo points on the circle

• Diameter: a chordcontaining the center

• Tangent: a line containing exactly one point of the circle Will be perpendicular to radius at that point

• Circumference:Length of theperimeter

• Sector:Pie shaped portionbounded by arc andtwo radii

Language of Circles

Language of Circles

• Segment:Region bounded byarc and chord

• Central angleCAB, center is theangle vertex

Language of Circles

• Inscribed angleCDB Vertex is on thecircle 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 theorema) P Q

b) If hypotheses then conclusion

3. Note the steps of Geogebra constructiona) 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 Qthen P Q

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

• Begin by assuming P and not Q• Use logical reasoning to look for contradiction• This gives us• Which means that P Q must be true

P Q P Q

P Q

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

• Also written

• Proof must proceed in both directions Assume P, show Q is true Assume Q, show P is true

P Q and Q P P Q

Proofs

• Constructed diagrams provide visual proof demonstration of geometric theorems

• Consider this diagram• How might it help

us prove that thenon adjacent angles ofa cyclic quadrilateralare supplementary.

Proof of Theorem 4.3

• Assume ABCD cyclic

• Consider pair of non-adjacentangles

• Let arc BAD be arc subtendingangle a, BCD be arc subtending b

• We know a + b = 360 and

• Also ½ a = , ½ b =

• So

• And they are supplementary

,BAD BCD

1( ) 1802a b

BAD BCD

180BAD BCD

Incircles and Excircles

• Consider concurrency of angle bisectors of exterior angles

• PerpendicularPJ gives radiusfor excircle

• Note the otherexterior anglesare congruent

• How to show tangency points M and N?

Incircles and Excircles

• Proof : Drop perpendiculars from P to lines XY andXZ

• Look for congruentright triangles

• Finish the proof

Families of Circles

• Orthogonal circles:Tangents are perpendicular at points of intersection

• Describe how youconstructed 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 allthe semicircles

• Calculate the are of circlewith diameter RP

• Show equality

Power of a Point

• We are familiar with the concept of a function Examples:

• Actually Geogebra commands are functions that take points and/or lines as parameters

2 3

2

( ) 6 9

( , ) 3 7 4

f x x x

g x y x xy y

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

or …

Given line intersecting circle O at Q and R with collinear point P use

2 2d r

PQ PR

Point at Infinity

• Recall stipulation Pcannot be at O

• As P approachesO, P’ gets infinitelyfar 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 GeometryRobust Constructions

Proofs

Chapter 4