Circles in Neutral Geometry Text Section 3.8 We Will Use the Usual

Circles in Neutral Geometry Text section 3.8

We will use the usual definition of circle - the one that leads to “square circles” in the Taxicab plane. Many of the familiar properties of circles in the Euclidean plane are inherited from neutral geometry - only those that somehow depend on uniqueness of parallels or a specific angle sum in triangles will not hold. This section gives an assortment of circle definitions and properties in neutral geometry.

Definition. A circle is the set of all points in a plane that lie at a fixed distance r from a fixed point O. The point O is the center and the distance r is the radius of the circle. [Note: the term radius is also applied to any segment with one endpoint at the center and the other on the circle, when there is no danger of confusion]. A point P is an interior point of the circle (interior to the circle or in the interior of the circle) if OP < r; P is an exterior point of the circle (exterior to the circle or in the exterior of the circle) if OP > r.

Informally, we may say an interior point of a circle is “inside the circle” and an exterior point is “outside the circle”. This must be used carefully — it can lead to confusion because an interior point is not “in the circle” according to the formal deﬁnition of “circle” (not in the set of points that is the circle). [Exterior points an interior points are not points of the circle in the usual sense].

For convenience in deﬁning a number of terms associated with circles, we will refer to the union of a line with either of its half-planes as a closed half-plane.

Deﬁnition. In any circle: A chord is a segment whose endpoints are on the circle. A diameter is a chord which passes through the center of the circle. A secant to the circle is a line which meets the circle at exactly two points . A tangent to the circle is a line which meets the circle at exactly one point (called the point of tangency or the point of contact. A central angle is an angle 6 AOB for which O is the center of the circle and A, B are on the circle. An arc of a circle is that part of a circle lying in a closed halfplane determined by a secant; the endpoints of the arc are the points of the arc lying on that secant. For three distinct points A, B, C on a circle, we use the notation ABCd to represent the arc with endpoints A and C which contains point B. [Note that naming an arc with just the endpoints would be ambiguous – there are two arcs with the same endpoints] An angle is inscribed in an arc if it can be named 6 ABC with A, C the endpoints of the arc and B another point of the arc. (Such an angle may also be called an “inscribed angle” without explicit reference to the arc) An arc or chord is subtended by an angle if the endpoints of the arc or chord are on the two sides of the angle and the other points of the arc or chord are in the interior of the angle.

Many of the well-known properties of circles in Euclidean geometry also hold in neutral geometry (because they don’t fundamentally involve parallelism):

• The center of a circle is the midpoint of any diameter.

• The perpendicular bisector of any chord of a circle passes through the center of the circle (follows from the ”equidistant” characterization of the perpendicular bisector).

• A line passing through the center of the circle and perpendicular to a chord of the circle bisects the chord (same reason).

• Two congruent central angles subtend congruent chords – and congruent chords are subtended by congruent central angles (SAS axiom).

1 • Two chords equidistant from the center of a circle have equal lengths, and two chords with the same length are the same distance from the center of the circle (HL congruence property with the ”perpendicular is bisector” properties, above - recall “distance from center to chord” is ”length of the perpendicular segment” since it is the distance from a point to a line). Kay shows proofs of a couple of these.

From the definitions given, we can see that every central angle and every diameter in a circle determines two arcs (the two arcs whose endpoints are on the sides of the angle or are endpoints of the diameter). We can also see that every arc determines a central angle or a diameter (the angle whose sides pass through the endpoints of the arc or the diameter whose ends are endpoints of the arc). This correspondence between {central angles and diameters} and {pairs of arcs} allows us to use central angles and diameters to classify arcs. Definition. Classification of arcs. An arc is a semicircle if it is the intersection of its circle with a closed halfplane determined by a diameter of the circle. An arc is a minor arc if it lies on and in the interior of its central angle. [If it is subtended by its central angle] An arc is a major arc if it is not a semicircle and does not lie on and in the interior of its central angle. Once again (as with segments and angles), it will be convenient to have a measure for arcs – to allow us to compare “size”. This measure can be defined in every neutral geometry – no new axioms are required to add it to our system: Definition. For any arc ABCd in a circle with center O, we define the measure mABCd of the arc by  m6 AOC if ABCd is a minor arc,  mABCd = 180, if ABCd is a semicircle,  360 − m6 AOC if ABCd is a major arc. Notice that arc measure is always positive and less than 360, unlike angle measure which is always less than 180 Theorem 1 (Additivity of arc measure). If APd B and BQCd are two arcs in a circle which meet only at B, and if their union ABCd = APd B ∪ BQCd is an arc of the circle, then mABCd = mAPd B + mBQCd . Theorem 2. (Tangent theorem) A line is tangent to a circle iff it is perpendicular to a radius at a point of the circle. ←→ ←→ Corollary. If two tangents PA and PB to a circle with center O have A and B as points of contact with −→ the circle, then PA =∼ PB and PO bisects 6 AP B. We would like to say that the interior of a circle is bounded by the circle – that any line that meets the interior of the circle must meet the circle itself. We would like to believe this is already built into our system and does not require another axiom, and this is the case, but we do need to go back to the ruler axiom to prove it, because we need the notions of continuity and completeness. If we did not have some sort of completeness axiom (the Ruler axiom lets us “borrow” the completeness of the real number system), we would need to take this theorem [Theorem 3] as an axiom. −→ ←→ Lemma 1. Let AB be any ray, O a point not on the line AB and f : AB → R a coordinate system on ←→ AB for which f(A) = 0 and f(B) > 0. If we define a function d : R → R by d(x) = OP for the point P with f(P ) = x, then d is a continuous function. The point of this lemma is that we can use the Intermediate Value Theorem to say that if there are −→ −→ points S and T on AB then, for any distance q with OS < q < QR, there is a point Q on AB with OQ = q. This is closely related to the idea of segment construction – but does not require a segment to be copied, and it does not involve marking off a distance on an already chosen ray (The point O is not on the ray −→ AB)

2 Theorem 3. (Secant theorem - or Line/Circle theorem). If a line l passes through a point A which is interior to a circle, then the line is a secant of the circle.

Corollary. Any segment joining a point interior to a circle to a point exterior to circle meets the circle.

This says, in essence, that a circle, like a line, divides the rest of the plane into two regions, and we have to cross the circle to get from one to the other. This would not be true in a non-ruler geometry such as the Rational Cartesian Plane (which was eliminated from our theory by the ruler axiom) in which the 2 2 line given by y = x contains the point (0, 0) inside the circle given by√x + y√ = 1 but does√ not actually√ meet the circle (only common solutions of y = x and x2 + y2 = 1 are ( 2/2, 2/2) and (− 2/2, − 2/2) - and these do not correspond to points of the Rational Cartesian Plane). In the circle case (unlike the line), the two regions are fundamentally different - one (the interior) is convex and the other (the exterior) is not. The interior has a additional very important property of being “bounded” (there is a finite upper bound on the distance between points in the set – this property is more general than the result in Theorem 3) which also distinguishes it from the exterior. This separation property generalizes to any simple closed curve (a curve which begins and ends in the same place and does not cross itself at any other point) but the proof is well beyond the scope of this course [At the least, we would need a working definition of “curve”]. Some of you have seen applications in Discrete Mathematics, the math majors will see the idea [including a working definition of “curve”] in Complex Analysis. Triangles, quadrilaterals, hexagons, ellipses, etc. are simple closed curves - as are shorelines of continents (which are too complicated to draw exactly – “simple” has a precise technical meaning which is distinct from “easy to visualize”), etc.