# TRIANGLE

, in Geometry, a figure bounded or contained by three lines or sides, and which consequently has three angles, from whence the figure takes its name.

Triangles are either plane or spherical or curvilinear. Plane when the three sides of the Triangle are right lines; but spherical when some or all of them are arcs of great circles on the sphere.

Plane Triangles take several denominations, both from the relation of their angles, and of their sides, as below. And 1st with regard to the sides.

An *Equilateral Triangle,* is that which has all its
three sides equal to one another; as A.

An *Isosceles* or *Equicrural Triangle,* is that which has
two sides equal; as B.

A *Scalene Triangle* has all its sides unequal; as C.

Again, with respect to the Angles.

A *Rectangular* or *Right-angled Triangle,* is that
which has one right angle; as D.

An *Oblique Triangle* is that which has no right angle,
but all oblique ones; as E or F.

An *Acutangular* or *Oxygone Triangle,* is that which
has three acute angles; as E.

An *Obtusangular* or *Amblygone Triangle,* is that which
has an obtuse angle; as F.

A *Curvilinear* or *Curvilineal Triangle,* is one that
has all its three sides curve lines.

A *Mixtilinear Triangle* is one that has its sides some
of them curves, and some right lines.

A *Spherical Triangle* is one that has its sides, or at
least some of them, arcs of great circles of the sphere.

*Similar Triangles* are such as have the angles in the
one equal to the angles in the other, each to each.

The *Base* of a Triangle, is any side on which a perpendicular
is drawn from the opposite angle, called the
*vertex;* and the two sides about the perpendicular, or
the vertex, are called the *legs.*

*The Chief Properties of Plane Triangles,* are as follow,
viz, In any plane Triangle,

1. The greatest side is opposite to the greatest angle, and the least side to the least angle, &c. Also, if two sides be equal, their opposite angles are equal; and if the Triangle be equilateral, or have all its sides equal, it will also be equiangular, or have all its angles equal to one another.

2. Any side of a Triangle is less than the sum, but greater than the difference, of the other two sides.

3. The sum of all the three angles, taken together, is equal to two right angles.

4. If one side of a Triangle be produced out, the external angle, made by it and the adjacent side, is equal to the sum of the two opposite internal angles.

5. A line drawn parallel to one side of a Triangle, cuts the other two sides proportionally, the corresponding segments being proportional, each to each, and to the whole sides; and the Triangle cut off is similar to the whole Triangle.

If a perpendicular be let fall from any angle of a Triangle, as a vertical angle, upon the opposite side as a base; then

6. The rectangle of the sum and difference of the sides, is equal to twice the rectangle of the base and the distance of the perpendicular from the middle of the base.—Or, which is the same thing in other words,

7. The difference of the squares of the sides, is equal to the difference of the squares of the segments of the base. Or, as the base is to the sum of the sides, so is the difference of the sides, to the difference of the segments of the base.

8. The rectangle of the legs or sides, is equal to the rectangle of the perpendicular and the diameter of the circumscribing circle.

If a line be drawn bisecting any angle, to the base or opposite side; then, |

9. The segments of the base, made by the line bisecting the opposite angle, are proportional to the sides adjacent to them.

10. The square of the line bisecting the angle, is equal to the difference between the rectangle of the sides and the rectangle of the segments of the base.

If a line be drawn from any angle to the middle of the opposite side, or bisecting the base; then

11. The sum of the squares of the sides, is equal to twice the sum of the squares of half the base and the line bisecting the base.

12. The angle made by the perpendicular from any angle and the line drawn from the same angle to the middle of the base, is equal to half the difference of the angles at the base.

13. If through any point D, within a Triangle ABC, three lines EF, GH, IK, be drawn parallel to the three sides of the Triangle; the continual products or solids made by the alternate segments of these lines will be equal; viz, .

14. If three lines AL, BM, CN, be drawn from the three angles through any point D within a Triangle, to the opposite sides; the solid products of the alternate segments of the sides are equal; viz, , (2d fig. above).

15. Three lines drawn from the three angles of a Triangle to bisect the opposite sides, or to the middle of the opposite sides, do all intersect one another in the same point D, and that point is the centre of gravity of the Triangle, and the distance AD of that point from any angle as D, is equal to double the distance DL from the opposite side; or one segment of any of these lines is double the other segment: moreover the sum of the squares of the three bisecting lines, is 3/4 of the sum of the squares of the three sides of the Triangle.

16. Three perpendiculars bisecting the three sides of a Triangle, all intersect in one point, and that point is the centre of the circumscribing circle.

17. Three lines bisecting the three angles of a Triangle, all intersect in one point, and that point is the centre of the inscribed circle.

18. Three perpendiculars drawn from the three angles of a Triangle, upon the opposite sides, all intersect in one point.

19. If the three angles of a Triangle be bisected by the lines AD, BD, CD (3d fig. above), and any one as BD be continued to the opposite side at O, and DP be drawn perp. to that side; then is [angle]ADO = [angle]CDP, or [angle]ADP = [angle]CDO.

20. Any Triangle may have a circle circumscribed about it, or touching all its angles, and a circle inscribed within it, or touching all its sides.

21. The square of the side of an equilateral Triangles is equal to 3 times the square of the radius of its circumscribing circle.

22. If the three angles of one Triangle be equal to the three angles of another Triangle, each to each; then those two Triangles are similar, and their like sides are proportional to one another, and the areas of the two Triangles are to each other as the squares of their like sides.

23. If two Triangles have any three parts of the one (except the three angles), equal to three corresponding parts of the other, each to each; those two Triangles are not only similar, but also identical, or having all their six corresponding parts equal, and their areas equal.

24. Triangles standing upon the same base, and between the same parallels, are equal; and Triangles upon equal bases, and having equal altitudes, are equal.

25. Triangles on equal bases, are to one another as their altitudes: and Triangles of equal altitudes, are to one another as their bases; also equal Triangles have their bases and altitudes reciprocally proportional.

26. Any Triangle is equal to half its circumscribing parallelogram, or half the parallelogram on the same or an equal base, and of the same or equal altitude.

27. Therefore the area of any Triangle is found, by multiplying the base by the altitude, and taking half the product.

28. The area is also found thus: Multiply any two sides together, and multiply the product by the sine of their included angle, to radius 1, and divided by 2.

29. The area is also otherwise found thus, when the three sides are given: Add the three sides together, and take half their sum; then from this half sum subtract each side severally, and multiply the three remainders and the half sum continually together; then the square root of the last product will be the area of the Triangle.

30. In a right-angled Triangle, if a perpendicular be let fall from the right angle upon the hypothenuse, it will divide it into two other Triangles similar to one another, and to the whole Triangle.

31. In a right-angled Triangle, the square of the hypothenuse is equal to the sum of the squares of the two sides; and, in general, any figure described upon the hypothenuse, is equal to the sum of two similar figures described upon the two sides.

32. In an isosceles Triangle, if a line be drawn from the vertex to any point in the base; the square of that line together with the rectangle of the segments of the base, is equal to the square of the side.

33. If one angle of a Triangle be equal to 120°; the square of the base will be equal to the squares of both the sides, together with the rectangle of those sides; and if those sides be equal to each other, then the square of the base will be equal to three times the square of one side, or equal to 12 times the square of the perpendicular from the angle upon the base.

34. In the same Triangle, viz, having one angle equal to 120°; the difference of the cubes of the sides, about that angle, is equal to a solid contained by the difference of the sides and the square of the base; and the sum of the cubes of the sides, is equal to a solid contained by the sum of the sides and the difference | between the square of the base and twice the rectangle of the sides.

There are many other properties of Triangles to be found among the geometrical writers; so Gregory St. Vincent has written a folio volume upon Triangles; there are also several in his Quadrature of the circle. See also other properties under the article Trigonometry.

For the properties of spherical Triangles, see SPHERICAL
*Triangles.*

*Solution of* Triangles. See Trigonometry.

, in Astronomy, one of the 48 ancient
constellations, situated in the northern hemisphere.
There is also the *Southern Triangle* in the southern hemisphere,
which is a modern constellation. The stars
in the Northern Triangle are, in Ptolomy's catalogue
4, in Tycho's 4, in Hevelius's 12, and in the
British catalogue 16.

The stars in the Southern Triangle are, in Sharp's catalogue, 5.

*Arithmetical* Triangle, a kind of numeral Triangle,
or Triangle of numbers, being a table of certain numbers
disposed in form of a Triangle. It was so called
by Pascal; but he was not the inventor of this table, as
some writers have imagined, its properties having been
treated of by other authors, some centuries before
him, as is shewn in my Mathematical Tracts, vol. 1,
pa. 69 &c.

The form of the Triangle is as follows:

And it is constructed by adding always the last two numbers of the next two preceding columns together, to give the next succeeding column of numbers.

The first vertical column consists of units; the 2d a series of the natural numbers 1, 2, 3, 4, 5, &c; the 3d a series of Triangular numbers 1, 3, 6, 10, &c; the 4th a series of pyramidal numbers, &c. The oblique diagonal rows, descending from left to right, are also the same as the vertical columns. And the numbers taken on the horizontal lines are the co-efficients of the different powers of a binomial. Many other properties and uses of these numbers have been delivered by various authors, as may be seen in the Introduction to my Mathematical Tables, pages 7, 8, 75, 76, 77, 89, 2d edition.

After these, Pascal wrote a treatise on the Arithmetical Triangle, which is contained in the 5th volume of his works, published at Paris and the Hague in 1779, in 5 volumes, 8vo.

In this publication is also a description, taken from
the 1st volume of the French Encyclopedie, art.
*Arithmetique Machine,* of that admirable machine in-
vented by Pascal at the age of 19, furnishing an easy
and expeditious method of making all sorts of arithmetical
calculations without any other assistance than
the eye and the hand.