The parts of parallels intercepted between parallels are equal, and reciprocally. Three parallels always divide any two right lines into proportional parts. The ratio of these parts may be incommensurable.-- Application to the case in which a right line is drawn, in a triangle, parallel to one of its sides. To find a fourth proportional to three given lines. The right line, which bisects one of the angles of a triangle, divides the opposite side into two segments proportional to the adjacent sides. _Of similar triangles._ Conditions of similitude.--To construct on a given right line, a triangle similar to a given triangle. Any number of right lines, passing through the same point and met by two parallels, are divided by these parallels into proportional parts, and divide them also into proportional parts.--To divide a given right line in the same manner as another is divided.--Division of a right line into equal parts. If from the right angle of a right-angled triangle a perpendicular is let fall upon the hypothenuse, 1º this perpendicular will divide the triangle into two others which will be similar to it, and therefore to each other; 2º it will divide the hypothenuse into two segments, such that each side of the right angle will be a mean proportional between the adjacent segment and the entire hypothenuse; 3º the perpendicular will be a mean proportional between the two segments of the hypothenuse. In a right-angled triangle, the square of the number which expresses the length of the hypothenuse is equal to the sum of the squares of the numbers which express the lengths of the other two sides. The three sides of any triangle being expressed in numbers, if from the extremity of one of the sides a perpendicular is let fall on one of the other sides, the square of the first side will be equal to the sum of the squares of the other two, _minus_ twice the product of the side on which the perpendicular is let fall by the distance of that perpendicular from the angle opposite to the first side, if the angle is _acute_, and _plus_ twice the same product, if this angle is _obtuse_. _Of polygons._ Parallelograms.--Properties of their angles and of their diagonals. Division of polygons into triangles.--Sum of their interior angles.--Equality and construction of polygons. Similar polygons.--Their decomposition into similar triangles.--The right lines similarly situated in the two polygons are proportional to the homologous sides of the polygons.--To construct, on a given line, a polygon similar to a given polygon.--The perimeters of two similar polygons are to each other as the homologous sides of these polygons. _Of the right line and the circumference of the circle._ Simultaneous equality of arcs and chords in the same circle.--The greatest arc has the greatest chord, and reciprocally.--Two arcs being given in the same circle or in equal-circles, to find the ratio of their lengths.