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  lineconsidered 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:

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

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 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

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

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

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 Let arc BAD be arc subtending angle a, BCD be arc subtending b We know a + b = 360  and Also ½ a =, ½ b = So 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 function  Examples: 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 or …  Given line intersecting circle O at Q and R with collinear point P use

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 d 2 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 (2 nd 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