Elements of Geometry and Trigonometry |
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Page 77
... equivalent to H E D B AB2 + BC2 , and there will evidently remain the square ACDE ; hence the theorem is true . Scholium . This proposition is equivalent to the algebraical formula , ( a - b ) 2 — a2 — 2ab + b2 . PROPOSITION X. THEOREM ...
... equivalent to H E D B AB2 + BC2 , and there will evidently remain the square ACDE ; hence the theorem is true . Scholium . This proposition is equivalent to the algebraical formula , ( a - b ) 2 — a2 — 2ab + b2 . PROPOSITION X. THEOREM ...
Page 78
... equivalent to the sum of the squares described on the other two sides . Let the triangle ABC be right angled at A ... equivalent to the square AH , which is double of the triangle HBC . In the same manner it may be proved , that the ...
... equivalent to the sum of the squares described on the other two sides . Let the triangle ABC be right angled at A ... equivalent to the square AH , which is double of the triangle HBC . In the same manner it may be proved , that the ...
Page 79
... equivalent to the square of the hypothenuse diminished by the square of the other side ; which is tus ex- pressed : AB2BC2 — AC2 . Cor . 2. It has just been shown that the square AH is equi- valent to the rectangle BDEF ; but by reason ...
... equivalent to the square of the hypothenuse diminished by the square of the other side ; which is tus ex- pressed : AB2BC2 — AC2 . Cor . 2. It has just been shown that the square AH is equi- valent to the rectangle BDEF ; but by reason ...
Page 81
... equivalent to the square described on the third ; for if the angle contained by the two sides is acute , the sum of their squares will be greater than the square of the opposite side ; if obtuse , it will be less . PROPOSITION XIV ...
... equivalent to the square described on the third ; for if the angle contained by the two sides is acute , the sum of their squares will be greater than the square of the opposite side ; if obtuse , it will be less . PROPOSITION XIV ...
Page 82
... equivalent ( Prop . II . Cor . 2. ) . The triangles ADE , BDE , whose common vertex is E , have the same altitude , and are to each other as their bases ( Prop . VI . Cor . ) ; hence we have ADE BDE :: AD : DB . : B D A The triangles ...
... equivalent ( Prop . II . Cor . 2. ) . The triangles ADE , BDE , whose common vertex is E , have the same altitude , and are to each other as their bases ( Prop . VI . Cor . ) ; hence we have ADE BDE :: AD : DB . : B D A The triangles ...
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Common terms and phrases
adjacent altitude angle ACB angle BAC ar.-comp base multiplied bisect Book VII centre chord circ circumference circumscribed common cone convex surface cosine Cotang cylinder diagonal diameter dicular distance divided draw drawn equally distant equations equivalent feet figure find the area formed four right angles frustum given angle given line gles greater homologous sides hypothenuse inscribed circle inscribed polygon intersection less Let ABC logarithm number of sides opposite parallelogram parallelopipedon pendicular perimeter perpen perpendicular perpendicular let fall plane MN polyedron polygon ABCDE PROBLEM proportional PROPOSITION pyramid quadrant quadrilateral quantities radii radius ratio rectangle regular polygon right angled triangle S-ABCDE Scholium secant segment similar sine slant height solid angle solid described sphere spherical polygon spherical triangle square described straight line tang tangent THEOREM triangle ABC triangular prism vertex
Popular passages
Page 19 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
Page 232 - ... the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator.
Page 11 - A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse.
Page 168 - The radius of a sphere is a straight line drawn from the centre to any point of the surface ; the diameter or axis is a line passing through this centre, and terminated on both sides by the surface.
Page 31 - Hence, the interior angles plus four right angles, is equal to twice as many right angles as the polygon...
Page 18 - America, but know that we are alive, that two and two make four, and that the sum of any two sides of a triangle is greater than the third side.
Page 20 - In an isosceles triangle the angles opposite the equal sides are equal.
Page 86 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. A D A' Hyp. In triangles ABC and A'B'C', To prove AABC A A'B'C' A'B' x A'C ' Proof. Draw the altitudes BD and B'D'.
Page 159 - S-ahc be the smaller : and suppose Aa to be the altitude of a prism, which having ABC for its base, is equal to their difference. Divide the altitude AT into equal parts Ax, xy, yz, &c. each less than Aa, and let k be one of those parts ; through the points of division...
Page 64 - To inscribe a circle in a given triangle. Let ABC be the given triangle. Bisect the angles A and B by the lines AO and BO, meeting at the point 0.