Equilateral triangle

“ By far the most important figure in architectural proportion, considered from the standpoint of geometry, is the equilateral triangle. It would seem that the eye has an especial fondness for this figure, just as the ear has for certain related sounds. Indeed it might not be too fanciful to assert that the common chord of any key (the tonic with its third and fifth) is the musical equivalent of the equilateral triangle. It is scarcely necessary to dwell upon the properties and unique perfection of this figure. ”

“ If you multiply the height of any equilateral triangle by the moiety of the basis upon which the line marking the height stands perpendicularly, the product gives the area of that triangle.
In every equilateral quadrangle, the product of one diameter multiplied by the moiety of the other will be equal to the area of it.
In any circle, the product of its diameter, multiplied by three and one-seventh, will be equal to the periphery. This is the rule generally followed in practical life, though it is not quite exact. The geometricians have two other methods. ”

“ 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. ”

“ For example, on the white background of the large equilateral triangle the small red equilateral triangle, which is a fourth of it, may be placed in such a way that each vertex is tangent to the middle of each side of the larger triangle.
There are also two squares, one of 7 centimeters on a side and the other 3.5. They have their respective frames with white backgrounds. The 7 centimeters square may be placed on the background of the 10 centimeters square in such a way that each corner touches the middle of each side of the frame. ”

“ This process can be shortened by omitting the squaring of the hypotenuse, and just dividing the square of the base by the hypotenuse, with the remainder then subtracted by the hypotenuse, as is known in algebraic math. Half of this remainder is then the length of the vertical, as is shown in this formula:
( (a^2) /a) - (b^2) = ( (b^2) /a) -a
Section 120 All this applies to a right triangle. If the triangle is not a right triangle but an equilateral triangle there is only a need to know the measure of one side, and there is no need to know any of the angles. ”

“ Often a dissection problem is quite easy apart from this limitation of pieces. At the time of the publication in the Weekly Dispatch, in 1902, of a method of cutting an equilateral triangle into four parts that will form a square (see No. 26, "Canterbury Puzzles") , no geometrician would have had any difficulty in doing what is required in five pieces: the whole point of the discovery lay in performing the little feat in four pieces only.
Mere approximations in the case of these problems are valueless; the solution must be geometrically exact, or it is not a solution at all. ”

“ These are the equilateral triangle, the square and the regular hexagon. It is well known to mathematicians that there is not a fourth way possible, in which a plane may be cut into little spaces that shall be equal, similar and regular, without leaving any interstices. ”

“ It may be observed that the two triangles in I. 8 are equal in all respects; Euclid however does not assert more than the equality of the angles opposite to the bases, and when he requires more than this result he obtains it by using I. 4.
I. 9. Here the equilateral triangle DEF is to be described on the side remote from A, because if it were described on the same, side, its vertex, F, might coincide with A, and then the construction would fail. I. II. The corollary was added by Simson, It is liable to serious objection. For we do not know how the perpendicular BE is to be drawn. ”

“ 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. ”

“ What more easy than for any one to look a little into his own thoughts, and there try whether he has, or can attain to have, an idea that shall correspond with the description that is here given of the general idea of a triangle—which is neither oblique nor rectangle, equilateral, equicrural nor scalenon, but all and none of these at once? ”

“ 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 falsehood 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 ”

“ It may often happen as a matter of fact that in some given matter a proposition of the form 'All B is A' is true simultaneously with 'All A is B.' Thus it is as true to say that 'All equiangular triangles are equilateral' as that 'All equilateral triangles are equiangular.' Nevertheless we are not logically warranted in inferring the one from the other. Each has to be established on its separate evidence. ”

“ This solid has 4 faces, 4 vertices and 6 edges. In a similar manner, four co vertical equilateral triangles stand on a square base. Two such sets placed base to base form the octahedron, which consequently has 8 faces, 6 vertices and 12 edges. Five equilateral triangles co vertically placed would stand on a pentagonal base, and it was found that, by forming several sets of such pyramids, a solid could be obtained which had 20 triangular faces, which met in pairs to form 30 edges, and in fives to form 12 vertices. ”

“ If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish. Hence in “trilinear” co-ordinates, with ABC as fundamental triangle, its equation is Pα + Qβ + Rγ = 0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this becomes the “line at infinity,” and the forces reduce to a couple.
The sum of the moments of the two forces of a couple is the same about any point in the plane. ”

“ Th. 32 (p. 55) is new. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then the intercepts on any other straight Line that cuts them are equal.' This is awkwardly worded (in fact, as it stands, its subject, as I pointed out in p. 193, is inconceivable) , and does not seem at all worth stating as a Theorem.
At p. 57 I see an 'Exercise' (No. 5) . 'Shew that the angles of an equiangular Triangle are equal to two-thirds of a right angle.' In this attempt I feel sure I should fail. ”

“ The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. ”

“ It is useless to consider here separately spherical trigonometry, which cannot give rise to any special philosophical consideration; since, essential as it is by the importance and the multiplicity of its uses, it can be treated at the present day only as a simple application of rectilinear trigonometry, which furnishes directly its fundamental equations, by substituting for the spherical triangle the corresponding trihedral angle. ”

“ This requires to divide the space about the centre into ten equal angles. Each will be 1/10th of a right angle, or 1/5th of two right angles. If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isosceles triangle, where the angle at the centre equals 1/5th of two right angles; hence each of the angles at the base will be 2/5ths of two right angles, as all three angles together equal two right angles. Thus we have to construct an isosceles triangle, having the angle at the vertex equal to half an angle at the base. ”

“ It will be observed that there are three figures corresponding to three cases which may exist; for the straight line drawn parallel to one side may cut the other sides, or may cut the other sides when they are produced through the extremities of the base, or may cut the other sides when they are produced through the vertex. In all these cases the triangles which are shewn to be equal have their vertices at the extremities of the base of the given triangle, and have for their common base the straight line which is, either by hypothesis or by demonstration, parallel to the base of the triangle. ”

“ Th. 17 (p. 30) is Euc. I. 26 (2nd part) proved by superposition instead of Euclid's method (which I prefer) of constructing a new Triangle.
Th. 18 (p. 32) is Euc. I. 17, with old proof.
Th. 19 (p. 33) is new. 'Of all the straight Lines that can he drawn from a given point to meet a given straight Line, the perpendicular is the shortest; and of the others, those making equal angles with the perpendicular are equal; and that which makes a greater angle with the perpendicular is greater than that which makes a less.' This I think deserves to be interpolated. ”

“ Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. ”

“ 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. ”

“ Christoph Clavius in 1574 deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight. John Wallis in 1663 showed that the postulate follows from the possibility of similar triangles on different scales. Girolamo Saccheri (1733) showed that it is sufficient to have a single triangle, the sum of whose angles is two right angles. Other equivalent forms may be obtained, but none shows any essential superiority to Euclid’s. Indeed plausibility, which is chiefly aimed at, becomes a positive demerit where it conceals a real assumption. ”

“ The Pattern (Fig. 277) .—A distance equal to the hypotenuse of triangle AD is set off upon any vertical line. These points are lettered A and 1. With point 1 as a center and a radius equal to the hypotenuse of triangle 1–B, an arc is drawn bearing away from point A. This is intersected by an arc drawn from point A, whose radius is equal to line AB of Fig. ”

“ If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the center of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij.
For a solid to be comprehended of right lines understand plaines comprehended of right lines, as in other places following. ”

“ Thus there is no conception of triangle in general. When the mind thinks of this figure, it always conceives one or more definite and particular triangles, which it accepts as representative of all possible or actual triangles. It thinks of one or of several triangles with a mental reservation, that the varieties of which that figure is susceptible are not exhausted by the specimens of which it is thinking. ”