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The Circle
A Mathematical Exploration beyond the Line
By Alfred S. Posamentier, Robert Geretschläger Prometheus Books
Copyright © 2016 Alfred S. Posamentier and Robert Geretschläger
All rights reserved.
ISBN: 978-1-63388-168-6
CHAPTER 1
CIRCLE RELATIONSHIPS IN BASIC PLANE GEOMETRY — AND AN EXTENSION
As we begin our exploration of the circle, the one aspect that most people recall is that the Greek letter π (pi) is somehow related to this important geometric shape. Most will recall that π represents the ratio of the circumference of a circle to its diameter. The two formulas that come instantly to mind are that the circumference of the circle is equal to 2πr, and that the area of the circle is equal to πr. In each case, r is the length of the radius of the circle. However, before we inspect the circle for its multitude of applications and appearances, we will review some of the basic essentials that you may have forgotten from your high-school geometry course.
First, we will review some basic terminology related to circles. We all know that the radius is the line segment joining the center of the circle to any point on the circle, while the diameter is the line segment joining two points on the circle and that also contains the center of the circle. A line segment joining two points on the circle is called a chord. A line is said to be tangent to the circle if it touches a circle at exactly one point; and a line that intersects the circle in two points is called a secant. Two more definitions to review: the area within a circle bounded by two radii and the arc between them is called a sector, and the area within a circle bounded by a chord and the arc of the circle is called a segment of the circle.
Some relationships useful to our study of the circle are the following:
• Any three non-collinear points determine a unique circle.
* The non-collinear points A, B, and C determine the unique circle O. (See figure 1.1.)
• The perpendicular bisector of a chord contains the center circle and the midpoints of the arcs it intersects.
* Line CD is the perpendicular bisector of AB; therefore, line CD contains the center of the circle and points C and D are the midpoints of arcs AB. (See figure 1.2.)
• A line perpendicular to a radius at the endpoint on the circle is tangent to the circle.
* Line AB is perpendicular to radius OC at point C and is therefore tangent to the circle at point C. (See figure 1.3.)
• From a point outside the circle, two tangents drawn to the circle have equal segments to the points of tendency.
* Tangents PA and PB are equal in length. (See figure 1.4.)
• A polygon is said to be inscribed in a circle if all of its vertices are on the circle.
* Polygon ABCDE is inscribed in circle O. (See figure 1.5.)
• A circle is said to be inscribed in a polygon if all of the polygon's sides are tangent to the circle. (See figure 1.6.)
* Circle O is inscribed in polygon ABCDE.
• If a secant segment and a tangent segment to the same circle share an endpoint in the exterior of the circle, then the length of the tangent segment is the mean proportional between the length of the secant segment and the length of its external segment.
* Tangent AP is the mean proportional between PC and PB, so that
PC/AP = AP/PB
(see figure 1.7).
• If two secant segments of the same circle share an endpoint in the exterior of the circle, then the product of the lengths of one secant segment and its external segment equals a product of the lengths of the other secant segment and its external segment.
* For the two secants PED and PBC, the following is true:
PD · PE = PC · PB. (See figure 1.8.)
• If two chords intersect in the interior of a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
* For the two chords intersecting at the point P, the following is true: PA · PB = PC · PD. (See figure 1.9.)
• A central angle is one formed by two radii of a circle, and it is equal in measure to its intercepted arc.
* The measure of angle AOB is equal to the degree measure of arc AB. (See figure 1.10.)
• The measure of an inscribed angle, one formed by two chords intersecting on the circle, is one half the measure of its intercepted arc.
* The measure of angle APB is one-half the measure of arc AB. (See figure 1.11.)
• The measure of an angle formed by a tangent and a chord of a circle is one half the measure of its intercepted arc.
* The measure of angle ABP is one-half the measure of arc AB. (See figure 1.12.)
• The measure of an angle formed by two chords intersecting at a point in the interior of the circle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
* The measure of angle BPD is equal to one-half the measure of the sum of the measures of arcs BD and AC. (See figure 1.9.)
• The measure of an angle formed by two secants of the circle intersecting at a point in the exterior of the circle is equal to one-half the difference of the measures of the intercepted arcs.
* The measure of angle DPC is one-half the measure of arc DC minus arc EB. (See figure 1.8.)
• The measure of an angle formed by a secant and a tangent to a circle, which intersect at a point in the exterior of the circle, is equal to one-half the difference of the measures of the intercepted arcs.
* The measure of angle APC is equal to one-half the measure of arc AC minus arc AB. (See figure 1.7.)
• The measure an angle formed by two tangents from a common external point to a circle is equal to one half the difference of the measures of the intercepted arcs. We also notice that the measure of the angle formed by two tangents from an external point is supplementary to the measure of the closer intercepted arc.
* The measure of angle APB is equal to the difference of the two arcs AB. Additionally, the measure of angle APB is supplementary to the closer arc AB. (See figure 1.4.)
This brief review of the essentials of circle relationships that are introduced in the high-school geometry course should provide the basic tools we will need as we investigate circles.
A NON-CIRCLE AS A CURVE OF CONSTANT BREADTH
Let's consider the circle from the basic point of view of its physical structure, namely, that it is a curve of constant breadth (i.e., diameter). That means that if it is placed between two parallel lines that are tangent to it, the circle can turn and will remain tangent to these two fixed parallel lines.
As we can plainly see in figure 1.13, the "breadth" of a circle is its diameter. Wherever we place the parallel tangents, they will always be the diameter-distance apart.
Curiously, the circle is not the only geometric shape that has this property. There is also a rather odd-looking shape called a Reuleaux triangle (see figure 1.14), which is named after the German engineer Franz Reuleaux (1829–1905), who taught at the Royal Technical University of Berlin, Germany. One might wonder how Franz Reuleaux ever thought of this triangle. It is said that he was in search of a button that was not round but still could fit through a button hole equally well from any orientation. His "triangle" solved the problem, as we will see in the following pages.
The Reuleaux triangle is formed by three circular arcs with equal radii and centers at each of the vertices of an equilateral triangle. It has many unusual properties, and it compares nicely to the circle of similar breadth. What do we mean by the "breadth" of the Reuleaux triangle? We refer to the distance between two parallel lines tangent to the curve (see figure 1.15) as the breadth of the curve. We now look carefully at the Reuleaux triangle and notice that no matter where we place these parallel tangents, they will always be the same distance apart — namely the radius of the arcs composing the triangle.
Before we inspect some of the fascinating properties of this Reuleaux triangle, such as the fact that it is analogous to the circle in its ratio of perimeter to breadth also equaling π we will discuss a "practical application" of the Reuleaux triangle.
If we were to try to turn a circular screw head (i.e., a screw that has no screwdriver slot on top) with a normal wrench (see figure 1.16), we would have no success. The wrench would slip and not allow a proper grip on the circular head of the screw. The same would hold true for a Reuleaux triangular head (figure 1.17). It, too, would slip, since it is a curve of constant breadth, just as the circle is.
When would this type of situation arise? During the summer months, it is commonplace for city kids to "illegally" turn on the fire hydrants to cool off on very hot days. Since the valve of the hydrant is usually a hexagon-shaped nut, they simply get a wrench to open the hydrant. If that nut were the shape of a Reuleaux triangle, then the wrench would slip along the curve just as it would along a circle. However, with the Reuleaux triangle-shaped nut, unlike a circle-shaped nut, we could have a special wrench with a congruent Reuleaux triangle shape that would fit about the nut and not slip. This would not be possible with a circular nut. Thus, the Fire Department would be equipped with a special Reuleaux wrench to open the hydrant in cases of fire, yet the Reuleaux triangle could protect against playful water opening and avoid the wasting of water in this manner. (Just as a matter of fact, the fire hydrants in New York City have pentagonal nuts, which also do not have parallel opposite sides and cannot be turned by a normal wrench.)
The Reuleaux triangle is said to be, like the circle, a closed curve of constant breadth. That is to say that when one measures the figure with calipers, it will have the same measure no matter where the parallel jaws of the calipers are placed. This is true for a circle and also for the Reuleaux triangle.
As we showed before, the Reuleaux triangle is formed by drawing circles, each centered at a different vertex of a given equilateral triangle, and each having a radius equal in length to the side of the equilateral triangle. (See figure 1.18.)
Surprisingly, the circumference of the Reuleaux triangle of the breadth d has exactly the same perimeter (i.e., circumference) as that of a circle with the diameter equal to the breadth of the Reuleaux triangle. The Reuleaux triangle (of breadth d) has a perimeter that equals
3 (1/6 (2πd)) =πd.
The circle with a diameter of length d has a circumference that is πd, which is the same as the perimeter of the Reuleaux triangle.
To see why the R euleaux triangle has the same ratio of perimeter to breadth as the circle — namely, π — we will consider the following. The perimeter is composed of three arcs, each 1/6 of a circle of radius, r. Therefore, the perimeter is
3 (1/6) (2πr) = π r.
Because the breadth is r, the ratio of perimeter to breadth is — πr,/r = π, which is exactly what we know about a circle — the ratio of its perimeter (i.e., circumference) to its breadth (i.e., diameter) is equal to π.
The comparison of the areas of these two figures is quite another thing. The areas are not equal. Let's compare them. We can find the area of the Reuleaux triangle in a clever way: add the three circle sectors that overlap in the equilateral triangle and then deduct the pieces that overlap (so that this region is actually only counted once and not three times).
The total area of the three overlapping circle sectors = 3 (1/6) (πr2.
The area of the equilateral triangle [MATHEMATICAL EXPRESSION OMITTED]
The area of the Reuleaux triangle = [MATHEMATICAL EXPRESSION OMITTED]
The area of a circle with diameter (of length r) = π (r/2)2 = πr2/4.
Comparing the areas of these two figures of equal breadth indicates that the area of the Reuleaux triangle is less than the area of the circle. This is consistent with our understanding of regular polygons, where the circle has the largest area for a given diameter. This was further generalized in 1915 by Austrian mathematician Wilhelm Blaschke (1885–1962), who proved that, given any number of such figures of equal breadth, the Reuleaux triangle will always possess the smallest area, and the circle will always have the greatest area. Also, the Reuleaux triangle has another interesting quality that stands in contrast to the circle.
We know a wheel rolls on a flat surface quite smoothly. If the Reuleaux triangle is "equivalent" to the circle, it, too, should be able to roll on a flat surface. Well, it can roll, but it wouldn't be a smooth roll because of the "pointed" corners. If furniture movers used a roller in the shape of a Reuleaux triangle instead of the usual circular cylinder roller, the furniture mover would not "bounce" the object being moved, but it would roll somewhat irregularly. Why is this? Notice that the center point (or centroid) of the rolling Reuleaux triangle will not stay at a constant parallel path to the surface being rolled on, as is the case for a circle. The end view of these rolling R euleaux triangles can be seen in figure 1.19.
We can make an adjustment to the Reuleaux triangle to give it rounded corners — without destroying its properties. If we extend the sides (length 5) of the equilateral triangle that was used to generate the Reuleaux triangle by an equal amount (say, a) through each vertex, and then draw six circular arcs alternately with the vertices of the triangle as centers (see figure 1.20), the result is a modified Reuleaux triangle with "rounded corners" that would allow for a smoother roll.
We now need to see that this modified Reuleaux triangle is of constant breadth and that the ratio of its perimeter to its breadth is π. (See figure 1.20.)
The sum of the lengths of the three smaller "comer arcs" is
3 (1/6) (2πa).
The sum of the lengths of the three larger "side arcs" is
3 (1/6 (2π)(s + a)).
The sum of the six arcs is π (s + a) + πa = π (s + 2 a). The breadth is (s + 2a), so the ratio of perimeter to breadth is π. When you would least expect it, π again shows up. In comparison, a circle with diameter (s + 2a) has a circumference of π(s + 2a), the same as the Reuleaux triangle.
Another astonishing property of the Reuleaux triangle is that a drill bit of the shape of a Reuleaux triangle could bore a square hole rather than the expected round hole. Or, to put this another way, the Reuleaux triangle is always in contact with each side of a square of appropriate size. This can be seen in figures 1.21 and 1.22. Remember, however, that this drill will not be rotating on a fixed axis; rather, the center of a Reuleaux triangle rotating in the square almost describes a circle; more exactly, it consists of four elliptical arcs. (The circle is the only curve of constant breadth, which has a balanced center of symmetry.)
English engineer Harry James Watt, who lived in Turtle Creek, Pennsylvania, recognized this in 1914, when he received a US patent (number 1241175) enabling these drills to be produced. The production of drills that can cut square holes was begun in 1916 by the Watt Brothers Tool Works in Wilmerding, Pennsylvania. Thus, the Reuleaux triangle can be rotated so that it always touches the sides of a square and thereby brushes over the sides of the square and also gets very close to the corners of the square. (Again, see figures 1.21 and 1.22.)
Felix Wankel (1902–1988), a German engineer, built an internal combustion engine for a car, which had an internal rotating part that was in the shape of a Reuleaux triangle and rotated in a chamber. It had fewer moving parts, and gave out more horsepower for its size than did the usual piston engines. The Wankel engine was first tried in 1957 and then put into production in the 1964 Mazda. Again, the unusual properties of the Reuleaux triangle made this type of engine possible.
There are lots of entertaining and useful ideas attached to this Reuleaux triangle, which is an analogue of the circle, and hence shares ownership of π with the circle.
Now armed with the basic tools of circle geometry and its related figures, we are ready to explore the many amazing relationships that lie ahead in our journey through the realm of the circle.
(Continues...)
Excerpted from The Circle by Alfred S. Posamentier, Robert Geretschläger. Copyright © 2016 Alfred S. Posamentier and Robert Geretschläger. Excerpted by permission of Prometheus Books.
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