More Geometry

Steps to develop practices, or skills and concepts. The development is based on using rule and patterns one at a time and one after another to get more rules and patterns, or more practices. The development illustrates the use of one way implication rules B if A (or you prefer, If A then B), one at a time and one after another. 

Steps and Topics:

• Step 1. Euclidean Geometry Leanly
• Step 2. Geometric Development of Real Numbers
• Step 3A. Complex Numbers Geometrically
• Step 3B.  Addition of Points, Arrows and Vectors in the Plane
• Step 4A - Unit Circle Trigonometry with links to vectors and complex numbers.
• Step 4B.   Radian Measure for Angles

Step 1. Euclidean Geometry Leanly
(Geometry without Coordinates) 

Euclid about 300 BC in his elements produced a codification of geometry based on the use of rules and patterns, one at a time, one after another. 

This area on Euclidean Geometry  on geometry before coordinates offers thought-based explanation of the following.  Try to read them in sequence.  

1. Correspondence between triangles. Here is an explicit definition, not always seen in class. 
2. Isometry of Triangles - Here is a definition.
3. Side-Side-Side (SSs) method for triangle construction and SSS like method for locating point.
4. Side Angle Side (SAS) method plus an application
   Ruler and Compass Construction to Bisect an Angle
5. Angle-Side-Angle (ASA)method, and ASA-like method for determining current location in navigation.
6. Isoceles  and Equilateral Triangles plus applications: Construction and Characterization of a Right Bisector of a Line Segment  and Ruler and Compass Construction of a Perpendicular from a Point to a line (with properties of such perpendiculars)
7. Side-Side-Side Failure 
8. SAS Failure or Near Failure 
9. ASA Failure - links with the parallel postulate
10. Parallel Lines - and angles associated with a transversal.
11. Triangle Angle Sum - from the parallel postulate
12. Similarity and Minimal Conditions for
13. Right Angle Trig., from Similarity
14. Trig & Similarity - More about the Connection
15. Parallelograms and their Properties
16. Kite Construction from triangles
17. Parallelogram Construction from triangles

Links:

1. Top Study Geometry:  Seven Interactive (step by steps) online proofs of (1) vertically opposite angles are equal, (2) Sum of angles in a triangle = 180 degrees (3) equality of angles at base of an isosceles triangle ..
2. TopStudy More Geometry More Seven More Interactive (step by steps) online proofs
3. Top Study MATH Link  Visit here for Arc, Area and Volume Calculation (Mensuration) formulas

Step 2. Geometric Development of Arithmetic with Signed (Real) Numbers 

Summary: This Geometric or Extrinsic Development of arithmetic methods for signed numbers begins with with Multiplication of Displacements (arrows or vectors)  by whole numbers and fractions first without signs, and then with  plus and minus signs as prefixes. Here multiplying by a positive number is the same as multiplying by an unsigned number, while multiplying by a negative number takes result as multiplying by an unsigned number and reverses it direction.    Next considers the collinear addition and further multiplication of  signed multiples of a single nonlinear  displacement - a so called unit displacement.   The addition of those multipliers implies rules for the addition of the multipliers or signed numbers. The multiplication of a multiply implies rules for multiplication of the multipliers.  Details follow. They imply axioms for real numbers, arithmetic with them included.  

Educational Note:  (a)  Assuming and explaining the axioms for real numbers provides an alternative to this step II  in geometry.  

Details (to be rewritten)

See too  Arithmetic Reference Page from whole numbers to real numbers. See the lesson links in the right hand column.

1. Addition of Collinear Movements: Define, then show this Addition is commutative. Then show identify repeated Addition of a Single Collinear with multiplication by whole numbers.  Then defined Multiplication of  Collinear Movements by proper and improper fractions - whole numbers and mixed numbers included. Finally,  extend that that multiplication to included multiplication by signed numbers.  Observe resultant of a head-to-tail sum of pair of collinear arrows has length equals  the sum of their lengths when the vectors have the same direction, and length equal to the difference when the vectors have opposite direction. Observe addition of displacement vectors or movements is commutative and associative.  The zero movement gives the additive identity property.  Each displacement has an negative, its additive inverse, a vector with the same length and the opposite direction. 
   
2. On a finite or infinite straight line, choose an origin and then use it to define position vectors for points in the line.   The addition of position vectors (signed numbers) is then defined by the head to tail addition of those collinear position vectors - possible in any order since addition of collinear displacements is commutative. The head to tail addition of position vectors is also associative. 
   
3. On a finite or infinite straight line, choose an origin and then use a unit vector (displacement), to defined signed coordinate for points on the line relative to the unit vector.  Each point may be identified with a position vector, the vector from the origin to itself. Each position vector is a signed number multiple of the unit vector. The addition of signed coordinates (signed numbers) is then defined or implied by the head to tail addition of those collinear position vectors - possible in any order since addition of collinear displacements is commutative. The head to tail addition of displacement is also associative. The identification of signed numbers with collinear movements (displacements) along a straight line thus defines addition, implies the effect of adding a zero displacement or zero; and implies the existence of additive inverses.   The previous discussion of multiplication of collinear displacements by signed numbers suggests how to multiply signed coordinates. 
   
4. Real Numbers: Signed numbers may be represented by proper and improper fractions, and by terminating or non-terminating decimals.  They may also be represented by square roots and arithmetic expressions.  And with coordinates on a straight line relative to an origin and a unit vector,  signed numbers may be identified with points on the line or their position vectors, and with a class of vectors of a given length and direction, equal modulo the location of their initial points. There-in an opportunity to introduce or name the real number line, and to identify key subsets of the real numbers: whole numbers, natural numbers, rational numbers and irrational numbers.
   
5. Signed Numbers: Arithmetic operations on signed numbers based on the addition and multiplication of collinear displacements: Signed numbers provide coordinates along the coordinate axes. They can be identified with displacements in the positive and negative direction along one of the axes - call it the horizontal axes.  Whence the rules for the addition, subtraction and multiplication of signed numbers follow from those for the addition, subtraction and multiplication of collinear vectors. The number of times one displacement is a multiple of another leads to the definition of division. 
   
   Remark: Now the commutative and associative property of addition of arrows implies the same for coordinates.  Teachers may tell students to assume them - give them as theorems with proofs available, but with proof mastery optional.  Formal discussion can be left to later. Focus on providing students with an operational command of arithmetic with signed numbers would be an option.  
   
6. Necessary Field Properties of Coordinates:  Commutative, Associative, Distributive Laws. Properties of 0 and 1. Use of Additive and Multiplicative Inverses in applying rules to subtraction and division. Products of non-zero factors are nonzero since product of nonzero unsigned numbers is nonzero, or since the area of a rectangle with sides > 0  is nonzero.  The distributive law is equivalent to a change of scale and direction of the unit vector for coordinates along an axis. 

Coordinates, relative or absolute?

Coordinates may be given relative to a choice of unit length and direction (a unit vector) along the coordinate axes of a map. Or, equivalently, coordinates may be given relative to a choice of unit length and a choice of positive direction for the coordinate axes. In both cases, these relative coordinates are ordered pairs of signed numbers.

Coordinates may also be given absolutely relative to a choice of unit length and choice of positive direction for each coordinate axis. For example, a point in a planar map may be determined by absolute coordinates

[⁺5 cm, ^-6 cm]

where here the unit of length is the centimeter cm. Implicit here (a first example) is the multiplication of the unit length by a signed number. Implicit here (another first example) is a multiplication of the unit and unit vectors along the coordinate axes by signed numbers.

Details and  Extra Theory
(some duplication of what is on the left
but with links to lessons). 

Addition of vectors (displacements) in the plane and more specifically collinear vectors in a line, their multiplication by signed numbers (coordinates) and their representation as signed numbers multiplies of a unit vector implies is or consistent with definition of the addition and multiplication of the signed number multipliers alone, apart from their role in representing collinear vectors as multiples of a given unit vector.  

• Unsigned Reals Numbers - use of unsigned decimals as coordinates.
  
• Signed Coordinates - Introduction of real numbers by prefixing signs to hitherto unsigned numbers.
  
• Plane Vectors - Navigation - use of arrows or vectors in describing piecewise linear paths in the plane; Head-to-tail addition; Associativity of in place head-to-tail addition.
  
• Horizontal Vectors & Adding Vector Multiples of unit vectors]. Addition of horizontal, more generally collinear, vectors that represent displacements, AND properties of this addition - commutativity included.
  

•  Adding Signed Numbers. The addition of signed numbers A and B is defined so the addition of multiples A and B of a  vector  equals the multiple A+B of the vector.

•  Multiplying Signed Numbers. The product or multiplication of signed numbers is defined so the multiplication by signed number A of a signed number multiple B  of a vector is equals the multiple AB of that vector.

• Distributive Law for Reals. The sum of collinear vectors given by multiples A and B of a nonzero k should not change if k = c m where m is another vector. Two methods of expressing the sum as multiple of c lead to the distributive property  (A+B)C = AC + BC for signed real numbers.

• [Real Numbers Axioms] The foregoing considerations imply a superset of the real number axioms assumed in modern mathematics curricula (or derived in a context free manner in pure mathematics.)

• Modular or Remainder Arithmetic for real numbers- Here is real number generalization of modular or remainder arithmetic for whole numbers. 
  

Step 3A. Complex Numbers Geometrically

All field properties of the complex numbers except for one, a distributive laws are simple consequences of  an extrinsic geometric definition of addition and multiplication of points in the plane using rectangular and polar coordinates, the assumption of the equivalent of the latter in determining points in the plane, and the field properties of real numbers.  Site coverage of arithmetic derives the latter field properties in an extrinsic (geometric)  manner  

Educational Notes: (a) Step or sub- step  6 below alone with the assumption of the distributive law for complex numbers and the axioms for real numbers would provide a short and quick  path for an operational command of complex numbers within an easily visualized geometric framework.  (b) Step I and II above are pre-requisites to substeps 3 to 5 below. 

This treatment (August 20, 2008)  provides a definition and properties via an application of Euclidean  geometry and the properties of real numbers. The aim is to prove the distributive law for multiplication over addition as all other field properties of the complex numbers are easy consequences of definitions, an assumed equivalent of rectangular and polar coordinates, and field properties of real numbers. Details follow in the next five lessons. 

1. Addition of points in the plane - Say or define how to compute sums and how the origin, the summands, and  the result provide the vertices of a quadrilateral with opposites sides equal in length (at least when the summands and origin are not collinear).
2. Multiplication of Points in the Plane. Say or define how to use the polar coordinates of a pair points in the plane to define a product. Introduce complex numbers and derive a few key formulas for the expression of products in terms of rectangular coordinates. Includes  notation for complex numbers and basic  rectangular coordinate, product formulas.
3. Distributive Law, Step I - - scaling distributes over addition. The proof here may employs similarity concepts
4. Distributive Law, Step II  -- rotation distributes over addition.  How rotation via angle distributes over addition. The proof here uses triangular isometry arguments. 
5. Distributive Law, Step III  rotation & scaling together distribute over addition. That gives a proof of the distributive law for complex numbers. It further implies product formulas in terms of real and imaginary parts.  
6. This simple introduction of complex numbers relies on the distributive law to complete the derivation and description of the field properties of complex numbers. Ignore references in it (written earlier) to the question of how to derive the distributive law.

The item 1 to 5 development  ( August 20, 2008) of the distributive law and how it implies a rectangular coordinate (real and imaginary part) formula for products  is preceded by several other developments in site pages of the complex numbers and their properties. The motivation for all developments comes from a simple description of physics as the addition and multiplication of arrows in the plane in several minutes of 1979 public presentation of the late Richard Feynman, a physicist second to none, and the observation that  he was describing complex numbers without mention of them in a wider presentation of his subject to a general audience. 

Remark (1):  The site area on Complex Numbers gives easy consequence for calculations involving unit circle trigonometry (see below) and vector  dot and cross-products in the plane - see below. Easy consequences include another proof of the Pythagorean theorem if the derivation of the distributive law, step I, relies on similarity theory for triangles. Easy consequences stem from the equality of two different ways to compute products, the first with polar, the second with rectangular.

Remark (2): Properties of complex numbers (algebraically described) make turn the simple to challenging  proof of many trig identities into routine algebraic simplification and comparison exercises.  College students of engineering and physics are often given the complex number viewpoint without any explanation. The foregoing embeds a complex number viewpoint into the development of geometry.

Remark (3) : Pages on complex numbers appear not only in the site area on complex numbers but also in the online volume Why Slopes & More Math, and in the site areas on 4. Euclidean Geometry, 5. Analytic Geometry/Functions and 6. Number Theory. To do: Provide a better guide to what is present, and fill-in missing gaps.

Step 3B: Addition of Points, Arrows and Vectors

Inform students that the Pythagorean theorem is a consequence of the 

1. Addition of Order Pairs, Subtraction of ordered pairs,  and Signed Number Multiples: These may be defined as follows:
   
   [a,b] + [c,d] = [a + c, b + d]
   
   [a,b] - [c,d] = [a - c, b - d]
   
   k [a,b] = [ka, kb]
   
   Teachers may identify points in the plane with the heads (terminal ends) of displacements from the origin, associate points in the plane with position vectors (tails at the origin) and then give vectorial diagrams to illustrate the previous operations.
   
2. Addition Parallelogram:  Isometry of Right triangles or the  The Pythagorean Theorem (see Chinese Square Dissection Proof) or  implies the origin [0,0], the addends [a,b] and [c,d], their sum [a,b] + [c,d] = [a + c, b + d] form the vertices of quadrilateral with opposites sides of equal length. 
   
   Uniqueness: The proof in Step II above that rotation distributes over addition can be modified to show that a parallelogram with vertices  at [0,0], [a,b], [c,d] and [u,v] with the latter opposite the origin implies [u,v] and [a,b] + [c,d] have the same polar coordinates, and hence must be equal. The charization of a parallelogram as quadrilateral with opposite sides equal.
   
3. Length of k[a,b]. The Similarity of Right triangles, or optionally, the Pythagorean Theorem (see Chinese Square Dissection Proof) or  implies the length R of the position vector of  k [a,b] = [ka, kb] is  |k| times the length r of the position vector of [a,b].  In brief  R = |k|r. 
   
   For why, see the proof in Step II above that scaling distributes over addition 
   
4. Distance r of a point [a,b] to the origin:   The Pythagorean Theorem (Chinese square or complex number proof) implies the formula  r = sqrt (a²+b²) for the distance r of the point [a,b] to the origin.
   
   The Pythagorean Formula also implies the formula d = sqrt ( [c- a]²+[d-b]²) for the distance d between two points [a,b] and [c, d] in the plane.  That formula implies the arrow from [a,b] to [c, d] and its additive inverse also have the length d.
   
5. Optional - Introduction of Dilatations that Fix the Origin:  The multiplication of points [a, b] by a number k > 0 gives  the image point k [a,b] = [ka, kb].  If the point [a, b] has distant r from the origin then the point [ka, kb] has distant kr from the origin.  
   
   Problems: Show if k is allowed to be a negative, the point [ka, kb] has distance |k|r from the origin.  Also show if d = sqrt ( [c- a]²+[d-b]²) gives the distance d between two points [a,b] and [c, d] in the plane, then the distance between image points [ka, kb] and [kc,kd] is |k|d.
   
6. Optional - Dilatations that fix the origin distribute over Point addition: The distributive law A(B+C) = AB +AC implies multiplying a points [a, b] by k > 0 and so multiply distances to the origin by k without changing direction distributes over addition of points in the plane: 
   
   k( [a,b] + [c,d])  = k [a + c, b + d] = [k(a+c), k(b+d)]  = [ka + kc, kb +kd] 
                             = [ka,kb] + [kc, kd]  = k [a,b] + k [c, d]
   
    
7. Coordinate Description of Arrows:  Arrows drawn in the plane are characterized by their initial and terminal points which in turn are characterized in a coordinate systems,  a pairs of order pairs [a,b] and [c,d].  That being said, drawn arrows may be characterized by providing the initial and tail coordinates [a,b] and describing the displacement via the difference [c-a, d - b] = [dx, dy] = head coordinates - tail coordinates.
   
8. Equality of Arrows Modulo Initial Position: Two drawn vectors are said to be identical, modulo tail position, when and only when the differences  of head and tail coordinates result in the same ordered pair [dx, dy].  It follows that vectors identical modulo tail position have equal lengths, are parallel and have the same direction. With the aid of coordinates, the converse can be implied.
   
   Comparison of Arrows:  Two arrows with the same length and same direction are said to be equal or identical modulo the location of their tails (initial points).  That situation occurs when the difference
   
   head coordinates - tail coordinates = [dx, dy] 
   
   is the same for both arrows. The position vector of the point with [dx, dy] determines an arrow.  All the arrows equal to this arrow, modulo position of initial points, have the same same length and direction as this position vector. 
   
   Note:  head coordinates = terminal point coordinates and tail coordinates = initial point coordinates
   
9. Position Arrow:   The position vector of  point [a,b] in the plane is the arrow which terminates at [a,b] and which has initial point at the origin [0,0].
   
10. What is a Vector:   Let [dx, dy] be a point in the plane.  The set of all arrows equal to the position vector of [dx, dy], modulo the location of initial point,  is said to be a vector.  Each element, an arrow, in the set (an equivalence class) is said to be an instance of the vector (equivalence class). All elements or arrows in the vector have same length and direction as position arrow of the point with rectangular coordinates [dx, dy]. 
    
    Advanced theory: Show addition of vectors using representatives is or can be well-defined. 
    
11. Appearance of a Parallelogram: Points with coordinates [a, b] and [c, d] may be identified with position vectors - tails at the origin. Then   
    
    [a,b] + [c,d] = [a + c, b + d]
    
    can be identified with the head to tail addition of the vector drawn from [a,b] to [a+c, b +d ]. The latter vector, modulo tail position, is identical with the vector [c, d]. Likewise the position vector associated with [a + c, b +d] is given by the head to tail sum of a vector from [c,d] to [a+c, b+d] with the position vector of [c, d] since the addition of coordinates is commutative. The vector [c,d] to [a+c, b+d] is equivalent, equal, identical to the position vector of [a,b], modulo position of initial points. 
    
    The parallelogram will be squashed (flattened) when the origin, [a,b] and [c,d] provide coordinates of collinear points.  The foregoing implies the sum of two points can be obtained by drawing a parallelogram, that obtained by taking the position vectors of points with coordinates  [a,b] & [c,d] as adjacent sides.

Step 4A.  Unit Circle Trigonometry with 
connection to complex numbers and vectors

This development links the unit circle approach to right triangle approach, and to the use of complex numbers. Properties of the latter simplify many explanations (proofs) involving trig and vectors.

1.  Unit Circle Trig  Trig with or on the Unit Circle.       The point coordinates (1,A) determine a point on the unit circle with angle A with respect to the horizontal (real) axis.  That point has rectangular coordinates [x,y] that depend on A.  It is clear that cos(A) and sin (A) are periodic functions with period 360 degrees. We write cos(A) = x and sin(A) = y.  Whence many many identities in the trig functions cos (A) and sin(A) follow from a comparison of polar-coordinate rule (add angles, multiple lengths) obtained expressions and rectangular coordinate (real and imaginary parts) expressions for products of complex numbers. The tangent function is then given by tan(A) = sin(A)/cos(A).  Reflections about the horizontal and vertical axises, and the 45 degree line y = x  implies  implies cos(A) is an even function and sin(A).   
   
   Remark: In the modern mathematics curricula I saw as a student, basic trig identities were established using the properties of real numbers and and geometric assumptions about rotations about the origin of a unit circle. The above explanation of how and why origin-fixing dilatations and rotations distribute over point addition uses properties of real numbers and like or equivalent geometric assumption.  The use of geometric assumptions about coordinates departs from the instrinsic viewpoint of pure mathematics to ease comprehension and involves an extrinsic or operational viewpoint of mathematics.   The instrinsic viewpoint can be developed in advanced college level courses that develop mathematics from axioms about sets (or other objects).
   
2. Connect to Right Triangle Trigonometry.    . Similarity implies the ratio of any two sides in a figure equals the ratio of the corresponding sides in any similar figure.  Hence for acute angles A,  cos(A) and sin(A) are given by the ratios of adjacent and opposite sides for the angle A to the unit length hypotenuse of a right triangle determined by angle A in the first quadrant.  That implies standard right triangle trig formulas for cos (A) = adjacent/hypotnuse and sin(A) = opposite/hypotenuse.  Likewise, tan (A) = opposite/adjacent. Now the isoceles right triangle with legs of length 1 and hypotenuse of length sqrt(2) can be use to calculate cos(A), sin(A) and  tan(A)  for A  = 45 degrees.  Further the equilateral triangle of sides 2, bisected in two by the right bisector of one of its 60 degree angles, an altitude, can be used to calculate cos(A),  sin (A) and tan(A) for A = 30 degrees and A = 60
    degrees.|
   
3. Trig function Values on the Unit Circle: For all multiples of 30 and 45 degrees in the range 0 to 360 degrees, and determine corresponding values of trig functions from their reflection induced algebraic properties. 
   
   Tabulating trig functions provides an alternative to drawing diagrams and solving problems through the use of similarity - next. 
   
4. Complex Numbers & Trig: See this site introduction of complex numbers, the easy consequences, and the connection to complex number, algebraic approach and derivation trig identities based on cis(A) = cos(A) + isin(A) = e^iA. The easy consequences include the Pythagorean theorem,  the cosine law and from it, a converse to the Pythagorean theorem: If the Pythagorean identity a²+b²=c² for three side lengths a, b and c of triangle then the triangle is a right triangle with hypotenuse of length c. The easy consequences also include formulas for dot and cross-products of points or arrows in the plane. 
   
5. Cosine Law: Give or derive this law as an easy consequence of the properties of complex numbers - two ways or multiply. Next apply the cosine law forwards, backwards and side ways to solve right triangles.  Point out the option of drawing similar triangles and measuring in each way the law is used. |
   
6. Trigonometric Identities: Use properties of complex numbers (two ways to multiply) to algebraically derive and verify trigonometric identities - the engineering way.  That aids and speeds the coverage of this topic.  In sum, start with complex number viewpoint - real and imaginary parts of exp(iq) and  show algebraic development and verification of trig identities using exp(iq)
   
7. Geometric Applications of Cross-Products in the plane:  Obtain  expressions for cross-products using trig and using rectangular coordinates as  another easy consequence of two ways to multiply complex numbers. . Show how to calculate area of a triangle, kites or parallelograms from SAS data. See the derivation and discussion of the sine law
   
8. Geometric Applications of Inner-Products:  Obtain  expressions for dot-products using trig and using rectangular coordinates as  another easy consequence of two ways to multiply complex numbers.  Show how to compute components of a vector - horizontal, vertical and in any direction.  
   
   Optional: Connect to force analysis in physic and phasor analysis in electricity.
   
9. Develop methods and formulas for construction,  Dimensions, Areas and Perimeters of Regular Polygons with the aid Roots of Unity and trigonometry

Step 4B. Angles in Radian Measure 

Similarity of Arcs and Sectors of Circles: Arcs and sectors of circles can be compared using their central angles and the radius of the circles containing the arc.  Two arcs or sectors are isometric if their radii and central angles are equal.  Two arcs or sectors appear to be similar if their central angles are equal.  Direct Measurement as well as map drawing  implies corresponding lengths in similar arcs are proportional.

Preparation for Calculus

1. Similar Sectors and Switch to Radians:  On a circle of radius r, the length of an arc subtended by a central angle A = n degrees is given by s = kn  where  k is a proportionality constant, and a backward use of the proportionality relation  s = kA when A = 360 degrees, implies 2pr = k 360 and hence k = pr/180. Now  
   
   s = k n = pr n/180  or s/r = pr n/180. 
   
   Whence the arclength the arc of a circle relative to its radius, in other words, the radian measure of the arc, is proportional to the central angle measure relative to degrees, and so is independent of the radius of the similar sectors of a circle - two sectors of different circles determined by a central sector being similar when and only when the central angles are equal. [Rewrite or clarify if need-be].  Give the radian measures exactly for all multiples of 15 degrees in the range 0 to 360 degrees. 
   
2. Trig function Values on the Unit Circle: Give the radian measures exactly for all multiples of 30 and 45 degrees in the range 0 to 360 degrees, and determine or review the corresponding values of trig functions. 

Remark: The use of radian measure simplifies formulas in calculus for derivatives of trig functions, starting with sine and cosine.