Elements of Geometry, Part 1Harper & Brothers, 1896 - Geometry |
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Page 18
... centres describe arcs with equal radii intersecting at Z. Join PZ . This will be the required per- pendicular . [ This can be proved correct after reaching § 104. ] PROPOSITION VII . THEOREM 36. If two straight lines are 18 PLANE GEOMETRY.
... centres describe arcs with equal radii intersecting at Z. Join PZ . This will be the required per- pendicular . [ This can be proved correct after reaching § 104. ] PROPOSITION VII . THEOREM 36. If two straight lines are 18 PLANE GEOMETRY.
Page 73
... radii and the intercepted arc . SECTOR The angle between the radii is called the angle of the sector . 3 * * These definitions are repeated from § 20 . 156. Def . - Concentric circles are circles which have BOOK II THE CIRCLE.
... radii and the intercepted arc . SECTOR The angle between the radii is called the angle of the sector . 3 * * These definitions are repeated from § 20 . 156. Def . - Concentric circles are circles which have BOOK II THE CIRCLE.
Page 74
... radii are equal , and if their centres be made to coincide they will coincide throughout ; conversely , equal circles have equal radii . C ' C - I. GIVEN - any two circles , Cand C ' 74 PLANE GEOMETRY.
... radii are equal , and if their centres be made to coincide they will coincide throughout ; conversely , equal circles have equal radii . C ' C - I. GIVEN - any two circles , Cand C ' 74 PLANE GEOMETRY.
Page 75
... radii would be unequal , which is con- trary to the hypothesis . Ax . 10 Therefore , the circumferences coincide , and the circles coincide and are equal . II . CONVERSELY : GIVEN TO PROVE two equal circles . their radii equal ...
... radii would be unequal , which is con- trary to the hypothesis . Ax . 10 Therefore , the circumferences coincide , and the circles coincide and are equal . II . CONVERSELY : GIVEN TO PROVE two equal circles . their radii equal ...
Page 76
... radii of equal circles . ] Then the arc AB will coincide with the arc A'B ' , and is equal to it . § 150 Q. E. D. II . CONVERSELY : GIVEN - equal circles having equal arcs AB and A'B ' . TO PROVE - the subtended angles O and O ' equal ...
... radii of equal circles . ] Then the arc AB will coincide with the arc A'B ' , and is equal to it . § 150 Q. E. D. II . CONVERSELY : GIVEN - equal circles having equal arcs AB and A'B ' . TO PROVE - the subtended angles O and O ' equal ...
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Common terms and phrases
ABCD adjacent angles altitude angles are equal angles of parallel apothem assigned quantity bisecting centre chord circumference circumscribed circle coincide decagon diagonals diameter distance divided Draw equal circles equilateral triangle Exercise exterior angle figure Find the area GEOMETRY given circle given line given point given straight line GIVEN TO PROVE given triangle GIVEN-the Hence homologous sides hypotenuse included angle intersection isosceles triangle length line parallel lines are parallel locus mean proportional measured by arc middle points number of sides opposite sides parallel axiom parallel to BC parallelogram perimeter PLANE GEOMETRY Q. E. D. PROPOSITION quadrilateral radii ratio of similitude rectangle regular inscribed regular polygon right angles right triangle segment similar polygons straight line joining tangent THEOREM third side third straight line triangle ABC triangle whose sides triangles are equal unequal vertex vertices
Popular passages
Page 248 - The area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.
Page 215 - The areas of two regular polygons of the same number of sides are to each other as the squares of their radii or as the squares of their apothems.
Page 63 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line ; it is required to draw a straight line through the point A, parallel to the straight hue BC. In BC take any point D, and join AD; and at the point A, in the straight line AD, make (I.
Page 49 - If, from a point within a triangle, two straight lines are drawn to the extremities of either side, their sum will be less than the sum of the other two sides of the triangle.
Page 148 - In any obtuse triangle, the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, increased by twice the product of one of these sides and the projection of the other side upon it.
Page 47 - ... the third side of the first is greater than the third side of the second.
Page 149 - Sines that the bisector of an angle of a triangle divides the opposite side into parts proportional to the adjacent sides.
Page 47 - If two triangles have two sides of one equal respectively to two sides of the other...
Page 100 - At a given point in a straight line to erect a perpendicular to that line. Let AB be the straight line, and let c D be a given point in it.
Page 140 - If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.