Chapter 2
Review of Related Literature and Studies
This study used a historical review of the literature to investigate the characteristics, properties and attributes of triangles. Related research of book publication, academic and mathematical journals from the fields of topology, geometry, number theory and analytic trigonometry were used as the primary source of literature review contained in this dissertation.
Related Studies
Several books, journals, magazines and unpublished materials related to this study were gathered and reviewed in this chapter.
To give a background in this study, Jim Wilson, EMT 725 presented that the area of the triangle can be seen as the sum of the areas of three triangular pieces with a common vertex at the incenter of the triangle. Thus,
Given a DABC with sides that have length a, b, and c.
Area =
=
Also, we all know that Perimeter of a triangle is equal to the sum of the sides of the triangle as illustrated below;
Given a triangle with sides that have length a, b, and c.
Perimeter = a + b + c.
For the form of all triangles, Euclid showed how the 3 types of triangles (Isosceles, Right and Scalene) are classified based on the relation of the sides of the radius of a circle. A careful person would note that this relationship does not necessarily mean triangles are generated inside a circle, only. On this page, he will show you just how clever Eulid was in hiding “his art”. It involves theorems 4 & 5, Book IV, “ the inscription and circumscription of all triangles”. These are famous and important theorems.
A hard and fast rule of geometry and logic is, “No theorem may be proved by a consequent theorem”. Thus, order of precedence is extremely important in mathematics. The first of these theorems, theorem 4 “ To draw a triangle around a circle such that each side of the triangle is touches the circle with any triangle ABC”. The theorem finds the center point p of the circle as a concurrence of angle bisectors. The radius of the circle is determined by constructing lines PD, PE and PF which are at right angles to each side of the triangle. Below is a diagram showing how this is all accomplished;
Figure 1 Figure 2 Figure 3
Generated theorems from the above figures:
Theorem A. Given a line tangent to a circle, the radius to the point of tangency is perpendicular to the tangent.
Theorem B. Two tangent segments drawn to a circle from a common point are congruent.
Theorem C. All radii of a circle are congruent.
The Inscribed Circle:
In any triangle, a circle can be inscribe in it. This circle will have center at the incenter (the point of concurrency of the angle-bisectors) and the radius of this circle is related to the area of the circle and the lengths of the sides by the following formula:
r= , where k is the area of the triangle and s is the semiperimeter.
Similarly, given any triangle, a circle can be circumscribed around the triangle, containing all three of the vertices. The center of this circle, called the circumcenter, is the point of concurrency of the perpendicular-bisectors of the sides. The radius R of this circle can be found as follows.
R= , where a, b and c are the lengths of the sides and k is the area of the triangle.
Radius of an inscribed circle:
The radius of a circle inscribed in a triangle is r=k/s, where k is the area of the triangle, and s is the semiperimeter of the triangle.
The area, k, of triangle ABC is the sum of three smaller triangles AOC, COB, and BOA.
The area of AOC is (1/2) br.
The area of COB is (1/2)ar.
The area of BOA is (1/2)cr.
So k=(1/2)(a + b + c) (r) = sr.
Heron’s Formula:
Given a triangle with sides that have lengths a, b, and c.
The semiperimeter s is half the sum of the lengths of the sides. S=a+b+c / 2
This time we will start with the area formula , where a and b are any two sides and C the included angle. We know that by the law of cosines. Thus we know that. Thus the area is
, where s is the semiperimeter.
Pythagorean theorem
The researcher also tries to determine the significance of Pythagorean theorem to the development of concepts about triangles. The Pythagorean theorem or Pythagoras’ theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, though the facts of the theorem were known before he lived. The theorem states:
The sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse.
(A right triangle is one with a right angle; the legs are the two sides that make up the right angle; the hypotenuse is the third side opposite the right angle; the square on a side of the triangle is a square, one of whose sides is that side of the triangle).
Since the area of a square is the square of the length of a side, we can also formulate the theorem as:
Given a right triangle, with legs of lengths a and b and hypotenuse of length c, then a2 + b2 = c2
Proof:
Draw a right triangle with sides a, b, and c as above. Then take a copy of this triangle and place its a side in line with the b side of the first, so that their c sides form a right angle (this is possible because the angles in any triangle add up to two right angles — think it through). Then place the a side of a third triangle in line with the b side of the second, again in such a manner that the c sides form a right angle. Finally, complete a square of side (a+b) by placing the a side of a fourth triangle in line with the b side of the third. On the one hand, the area of this square is (a+b)2 because (a+b) is the length of its sides. On the other hand, the square is made up of four equal triangles each having area ab/2 plus one square in the middle of side length c. So the total area of the square can also be written as 4 · ab/2 + c2. We may set those two expressions equal to each other and simplify:
a2 + 2ab + b2 = 2ab + c2
a2 + b2 = c2
Note that this proof does not work in non-Euclidean geometries, since, say, on a sphere, the angles of a triangle don’t add up to 180 degrees, and the above “square” cannot be formed. (See the external links below for a sampling of the many different proofs of the Pythagorean theorem.)
The converse of the Pythagorean theorem is also true:
For any three positive numbers a, b, and c such that a2+b2=c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b.
This can be proven using the law of cosines which is a generalization of the Pythagorean theorem applying to all (Euclidean) triangles, not just right-angled ones. Another generalization of the Pythagorean theorem was already given by Euclid in his Elements:
If one erects similar figures on the sides of a right triangle, then the sum of the areas of the two smaller ones equals the area of the larger one.
The Pythagorean theorem stated in Cartesian coordinates is the formula for the distance between points in the plane — if (a, b) and (c, d) are points in the plane, then the distance between them is given by
This distance formula generalizes to inner product spaces, and the version of the Pythagorean Theorem in inner product spaces is known as Parseval’s identity.
The Pythagorean theorem also generalizes to higher-dimensional simplexes. If a tetrahedron has a right angle corner (a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This is called de Gua’s theorem.
Since the Pythagorean theorem is derived from the axioms of Euclidean geometry, and physical space may not always be Euclidean, it need not be true of triangles in physical space. One of the first mathematicians to realize this was Carl Friedrich Gauss, who then carefully measured out large right triangles as part of his geographical surveys in order to check the theorem. He found no counterexamples to the theorem within his measurement precision. The theory of general relativity holds that matter and energy cause space to be non-Euclidean and the theorem does therefore not strictly apply in the presence of matter or energy. However, the deviation from Euclidean space is small except near strong gravitational sources such as black holes. Whether the theorem is violated over large cosmological scales is an open problem of cosmology.
Given two vectors, A and B, the Pythagorean Theorem states
||A + B||2 = ||A||2 + ||B||2 if and only if the two vectors are orthogonal.
Credit:ivythesis.typepad.com
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