Unidad 1: À mon avis
5.8 Sección: Documents p.19
The Ford circle of a rational number r = pq is the circle of radius 2q12 in the upper half plane tangent to the x-axis at the point (r, 0).
The Ford circles of two distinct rational numbers are either disjoint or tangent to each other externally. In the latter case we say that two rational numbers are adjacent.
Theorem 4.1. Two rational numbers are adjacent if and only if they are consecutive terms in a Farey sequence.
Corresponding to each Farey sequence, there is a sequence of tangent Ford circles. Here are the Ford circles corresponding toF5.
0 1
5 1 4 1
3 2
5 1
2 3
5 2 3 3
44
5 1
Exercise
1. If pq and PQ are adjacent rational numbers, what is the point of tan-gency of their Ford circles?
128 Basic geometric constructions
Appendix: Some basic construction principles
Theorem 4.2 (Perpendicular bisector locus). Given two distinct A and B on a plane, a point P is equidistant from A and B if and only if P lies on the perpendicular bisector of the segment AB.
Theorem 4.3 (Angle bisector locus). A point P is equidistant from two given intersecting lines if and only if it lies on the bisector of an angle between the two lines.
Note that two intersecting lines have two angle bisectors.
Theorem 4.4. If two circles are tangent to each other, the line joining their centers passes through the point of tangency.
The distance between their centers is the sum (respectively difference) of their radii if the tangency is external (respectively internal).
The Geometer’s Sketchpad
Basic commands
• Construct points, segments, lines, circles.
• Construct a segment, ray, or line through two specified points.
• Construct a circle given its center and a point on it. Construct a circle given its center and radius.
• Construct the intersection of two objects.
• Construct the midpoint of a segment.
• Given a line and a point P , construct lines perpendicular and par-allel to through P .
• Translate, rotate, and dilate.
• Hide an object. (If the sketch you are making becomes too compli-cated, you may wish to hide some of the objects to make it simpler.
Note that this is not the same as “delete”).
• Label an object. Change the name of the object.
• Save a Sketch.
• Create New Tool.
Toolbox
Open anew sketch. Selecta segment and build an equilateral triangle on it. Hide everything except the vertices and sides of the equilateral triangle. Selecteverything by dragging the mouse from top left to bot-tom right. Select Create New Tool, and type a name for the tool, say, equilateral triangle. A new tool will now appear.
Savethis as a filebasic shapes.gspin the foldertool folder.
From theFile menu, chooseDocument options,Add page,Blank page. On this blank page, construct a square on a segment. Select Create new tooland name thissquare. Savethe file.
You can extend thisbasic shapes.gsp tool by opening new blank pages and creating new tools. Remember tosavethe file.
The current basic shape.gsp file contains the following tools: equi-lateral triangle, square, rectangle, right triangle, parallelogram, rhombus.
You may of course add your own tools to the same file.
130 Basic geometric constructions
Exercise
1. Given triangle ABC, construct the equilateral triangles BCX, CAY and ABZ externally on the sides of the triangle. Join AX, BY , CZ. What can you say about the intersections, lengths, and direc-tions of these lines (segments)?
2. Show that the 90◦ angle of a right triangle is bisected by the line joining it to the center of the square on the hypotenuse.
3. Make a sketch to show that for two given positive quantities a and b,
a + b 2 ≥√
ab≥ 2ab a + b. 4. Construct the following diagram.
A B
C D
5. Construct the following diagram.
A B
C D
6. Two congruent circles of radii a have their centers on each other.
Consider the circle tangent to one of them internally, the other ex-ternally, and the line joining their centers. It is known that this circle has radius√43a. Construct the circle.
7. An equilateral triangle of side 2a is partitioned symmetrically into a quadrilateral, an isosceles triangle, and two other congruent trian-gles. If the inradii of the quadrilateral and the isosceles triangle are equal, the common inradius is (√
3−√
2)a. Construct the partition.
8. Outline a simple procedure to divide a segment into 123 equal parts.
Chapter 5
Greatest common divisor
1 gcd(a, b) as an integer combination of a and b 2 Nonnegative integer combinations of a and b 3 Cassini formula for Fibonacci numbers
4 gcd of generalized Fibonacci and Lucas numbers Appendix: The Euler φ-function
Exercise Project
k rk qk xk yk
−1 Fn+1 ∗ ∗ ∗ 1 0
0 Fn 1 0 1
1 Fn−1 1 F1 −F2
2 Fn−2 1 −F2 F3
3 Fn−3 1 F3 −F4
...
n− 3 F3 1 (−1)n−2Fn−3 (−1)n−1Fn−2
n− 2 F2 1 (−1)n−1Fn−2 (−1)nFn−1 n− 1 F1 1 (−1)nFn−1 (−1)n+1Fn
n 0 ∗ ∗ ∗
5.1 gcd(a, b) as an integer combination of a and b
It is well known that the gcd of two (positive) integers a and b can be calculated efficiently by repeated divisions. Assume a > b. We form two sequences rkand qkas follows. Beginning with r−1 = a and r0 = b, These divisions eventually terminate when some rndivides rn−1. In that case, gcd(a, b) = rn.
If, along with these divisions, we introduce two more sequences (xk) and (yk) with the same rule but specific initial values, namely,
xk+1 =xk−1− qkxk, x−1 = 1, x0 = 0;
yk+1=yk−1− qkyk, y−1 = 0, y0 = 1.
then we obtain gcd(a, b) as an integer combination of a and b: 1 gcd(a, b) = rn = axn+ byn.
Theorem 5.1. Given relatively prime integers a > b, there are unique integers h, k < a such that ak− bh = 1.
Proof. Clearly, xnand ynare opposite in sign. Take (k, h) = (xn,−yn) or (b + xn, a− yn) according as xn> 0 or < 0.
Corollary 5.2. Let p be a prime number. For every integer a not divisible by p, there exists a positive integer b < p such that ab− 1 is divisible by p.
1In each of these steps, rk= axk+ byk.