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angle contained angle subtended bisect a given centre or circumference CHANCERY LANE chords Cor.—Hence cumference cutting line describe a circle diagonal divide a given equal angles reciprocally equal arches equal circles equal sides equal to four equi equilateral and equiangular EUCLID'S ELEMENTS external segment finite straight line four straight lines given circle given finite straight given point given rectilinear angle given rectilinear figure given straight line given triangle half the line homologous sides inscribe an equilateral internal angles isosceles triangle line be bisected line be divided lines be drawn lines be proportional obtuse angle opposite angle parallelogram equal perpendicular point of contact points of section polygons PROB PROP rectangle contained remaining angles right angles side opposite side subtending similar and similarly Similar triangles spectively equal square of half straight line drawn straight line touches tangent THEOR three given straight touches a circle twice the rectangle whole line XVIII XXXIII
Seite 62 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Seite 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.
Seite 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.
Seite 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...
Seite 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.
Seite 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.
Seite 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...
Seite 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.
Seite 40 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.