Isosceles triangle

“ In a case therefore like the present, where the base of our isosceles triangle is to the other two sides as eight units to twelve thousand, it is impossible not to perceive that it behoves us to be singularly diffident as to the conclusion at which we have arrived, or rather it behoves us to take for granted that we are not unlikely to fall into the most important error. We have satisfied ourselves that the sides of the triangle including the apex, do not form an angle, till they have arrived at the extent of ninety-five millions of miles. ”

“ Conceive the isosceles triangle ABC to be taken up, and then replaced so that AB falls on the old position of AC, and AC falls on the old position of AB. Thus, in the manner of I. 4, we can shew that the angle ABC is equal to the angle ACB.
I. 6 is the converse of part of I. 5. One proposition is said to be the converse of another when the conclusion of each is the hypothesis of the other. Thus in I. 5 the hypothesis is the equality of the sides, and one conclusion is the equality of the angles ”

“ Draw an isosceles triangle whose equal sides shall each be less than the third side.
Write its two names within it.
Draw an oblique straight line twice as long as any short measure or unit.
At one end draw a straight line perpendicular to it, and three times as long as the same measure. ”

“ The three corners of the triangle are joined by three lines, each line made of a series of points which should symbolize each a linear series of cognate forms, and we have these three forms in the trapezohedron d, the three-faced octahedron e, and the four-faced cube f, each linking by a single unbroken series the three corner forms. These are placed each on its proper line on the diagram. They are each twenty-four-sided figures. In two, each side is an isosceles triangle, in one, a trapezium, a combination of two isosceles triangles. ”

“ But this rule applies, not always to the Tradesman, and still less often to the Soldiers, and to the Workmen; who indeed can hardly be said to deserve the name of human Figures, since they have not all their sides equal. With them therefore the Law of Nature does not hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles still. Nevertheless, all hope is not shut out, even from the Isosceles, that his posterity may ultimately rise above his degraded condition. ”

“ And, according to the most approved astronomical observations hitherto made, I have an isosceles triangle, eight thousand miles broad at its base, and ninety-five millions of miles in the length of each of the sides reaching from the base to the apex.
It is however obvious to the most indifferent observer, that the more any triangle, or other mathematical diagram, falls within the limits which our senses can conveniently embrace, the more securely, when our business is practical, and our purpose to apply the result to external objects, can we rely on the accuracy of our results. ”

“ When you hit upon the right one and leave it about, you'll start seeing things sooner or later. But your chances are small. It is not an isosceles triangle, but a scalene. The original pentagram was a very rough effort, far from symmetrical, and only by a fluke did it contain this dangerous triangle.
"How did I discover all this? It began with my investigation of the haunting of a cottage in Norfolk. I connected the phenomena with a small glass prism which had been lying about the place (the former occupant was a spectroscopist—until he went mad and was put away) . ”

“ If A be the vertex and BD the base of the constructed triangle in IV. 10, D being one of the two points of intersection of the two circles employed in the construction, and E the other, and AE be drawn meeting BD produced at G shew that GAB is another isosceles triangle of the same kind.
330. In the figure of IV. 10 if the two equal chords of the smaller circle be produced to cut the larger, and these points of section be joined, another triangle will be formed having the property required by the proposition. ”

“ Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important to know that the lines drawn from the extremities of the base of an isosceles triangle to the middle points of the opposite sides are equal! ”

“ The bases of two or more triangles having a common vertex are given, both in magnitude and position, and the sum of the areas is given; prove that the locus of the vertex is a right line.
30. If the sum of the perpendiculars let fall from a given point on the sides of a given rectilineal figure be given, the locus of the point is a right line.
31. ABC is an isosceles triangle whose equal sides are AB, AC; B′C′ is any secant cutting the equal sides in B′, C′, so that AB′ + AC′ = AB + AC: prove that B′C′ is greater than BC.
32. ”

“ Where one edge does not thus overreach, the tables of earth, which at a small distance resemble masses of ice, are raised up so that their faces form an isosceles triangle, leaving a cavity beneath. About the extremities of the transverse axis, is also an overlapping of two feet, which continues nearly two rods on the curve each way from the axis, and in most places is double, overreaching internally and externally, exhibiting likewise, some irregularity where the compressing forces acted at right angles to each other. ”

“ If there are no such things as isosceles triangles, Euclid's fifth Proposition is only formally true, like a theorem concerning the fourth dimension of space: merely consistent with his other assumptions. But if there be any triangles only approximately isosceles, the proof applies to them, making allowance for their concrete imperfection: the nearer their sides approach straightness and equality the more nearly equal will the opposite angles be. ”

“ Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of an individual ”

“ Without any knowledge of the kinds of triangles the student discovers by examining the various shapes of many triangles that there is a group in which none of the sides are equal. For the lack of a better name he terms these non-equilateral (scalene) . Further observation discloses another group in which two of the sides are equal. These he names bi-equilateral (isosceles) . Finally a third group is designated as tri-equilateral (equilateral) . This process is classification. Division would consist in separating the genus triangle into the three kinds—scalene, isosceles, equilateral. ”

“ To which I answer, that though the idea I have in view, whilst I make the demonstration be, for instance, that of an isosceles rectangular triangle, whose sides are of a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles of what sort or bigness soever. And that because neither the right angle nor the equality nor the determinate length of the sides are at all concerned in the demonstration. ’Tis true, the diagram I have in view includes all these particulars, but then there’s not the least mention made of them in the proof of the proposition. ”

“ Chapter 23 From what has been shown it appears plain that if one and the same thing is inherent in two, for instance, A in C and in D, when one is not predicated of the other, either not at all or not universally, then it is not always inherent according to something common. Thus to the isosceles and to the scalene triangle, the possession of angles equal to two right, is inherent according to something common, for it is inherent so far as each is a certain figure, and not so far as it is something else. ”

“ Thus should we mention the word triangle, and form the idea of a particular equilateral one to correspond to it, and should we afterwards assert, that the three angles of a triangle are equal to each other, the other individuals of a scalenum and isosceles, which we overlooked at first, immediately crowd in upon us, and make us perceive the falshood of this proposition, though it be true with relation to that idea, which we had formed. If the mind suggests not always these ideas upon occasion, it proceeds from some imperfection in its faculties ”

“ Rectilinear figures cannot be divided into triangles and quadrilaterals because there are rectilinear figures which have more than four sides. On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist. ”

“ For the nature of these Ideas is such, that the Mind of man cannot possibly deny but that they are such and such distinct Ideas, and that such and such affections belong unto them. As for example, That every Triangle is either Isopleuron, Isosceles, or Scalenum ”

“ Harmonic conjugate of the point at infinity. It is natural to inquire what position of B corresponds to the infinitely distant position of D. We have proved (§ 27) that the particular quadrangle K, L, M, N employed is of no consequence. We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem. We choose the point L so that the triangle ALC is isosceles (Fig. 7) . Since D is supposed to be at infinity, the line KM is parallel to AC. Therefore the triangles KAC and MAC are equal, and the triangle ANC is also isosceles. ”

“ If instead of two parallel series of right-angle triangles there are isosceles triangles, they may be known as alternate isosceles triangles. These triangles, when opposite, form a series of hour-glass figures or squares. This form is commonly accompanied by a row of dots, affixed to top and base, known as the dotted square or hour-glass figure. Hour-glass designs are commonly represented upright, but the angles of the triangles may be so placed that the series is horizontal, forming a continuous chain. ”

“ II It is a common custom to refer to the usual complication between one man and two ladies, or one lady and two men, or a lady and a man and a nobleman, or—well, any of those problems—as the triangle. But they are never unqualified triangles. They are always isosceles—never equilateral. So, upon the coming of Nevada Warren, she and Gilbert and Barbara Ross lined up into such a figurative triangle; and of that triangle Barbara formed the hypotenuse. ”

“ The various triangles differ greatly in their capabilities of producing Life forms, the equilateral and the obtuse isosceles being especially deficient in this regard and requiring to be combined with the other tablets. The fact that both the right isosceles and right scalene triangles produce Life forms in great variety seems to prove that, as Goldammer says, "the right angle predominates in the products of human activity." ”

“ Our Women are Straight Lines.
Our Soldiers and Lowest Classes of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short (often not exceeding half an inch) that they form at their vertices a very sharp and formidable angle. Indeed when their bases are of the most degraded type (not more than the eighth part of an inch in size) , they can hardly be distinguished from Straight Lines or Women; so extremely pointed are their vertices. With us, as with you, these Triangles are distinguished from others by being called Isosceles ”

“ Now of the two triangles, the isosceles has one form only; the scalene or unequal-sided has an infinite number. Of the infinite forms we must select the most beautiful, if we are to proceed in due order, and any one who can point out a more beautiful form than ours for the construction of these bodies, shall carry off the palm, not as an enemy, but as a friend. ”