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angle contained angle equal angle subtended base BOOK centre chords construct cutting cutting line D B PROP describe diagonal diameter double drawn E PROP equal circles equal sides equal to four equilateral and equiangular external external angle extremes F PROP falling G B PROP given circle given point given rectilinear angle given straight line given triangle greater greatest half the line homologous sides inscribe a circle inscribe an equilateral internal intersect less line be divided line equal lines be drawn lines be proportional mean meeting nearer opposite angle opposite sides parallel parallelograms pass perpendicular point of contact points of section polygons PROB produced ratio rectangle contained right angles segment side opposite similar similarly described spectively equal square of half stand straight line drawn tangent THEOR twice the rectangle unequal VIII whole line XXXIII
Side 62 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Side 29 - In every triangle, the square of the side subtending any of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.
Side 57 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Side 55 - IF an angle of a triangle be bisected by a straight line, which likewise cuts the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the...
Side 58 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Side 26 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.
Side 54 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally ; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...
Side 56 - The sides about the equal angles of equiangular triangles are proportionals ; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios Let ABC, DCE, be equiangular triangles, having the angle ABC equal to the angle DCE, and the angle ACB to the angle DEC, and consequently (4.
Side 40 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.