A Text-book of Euclid's Elements for the Use of Schools, Book 1 |
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Page 29
... angle ACB by the straight line CD , meeting AB at D. Then shall AB be bisected at the point D. Proof . For in the ... vertical angle of an isosceles triangle , also bisects the base . 2 . On a given base describe an isosceles triangle ...
... angle ACB by the straight line CD , meeting AB at D. Then shall AB be bisected at the point D. Proof . For in the ... vertical angle of an isosceles triangle , also bisects the base . 2 . On a given base describe an isosceles triangle ...
Page 33
... angle XAY is equal to the angle XBY ; and if XY be joined , that the angle AXY is equal to the angle BXY . 6. Shew ... vertical angle BAC ; ( iii ) AO , if produced , bisects BC at right angles . 13. Describe an isosceles triangle ...
... angle XAY is equal to the angle XBY ; and if XY be joined , that the angle AXY is equal to the angle BXY . 6. Shew ... vertical angle BAC ; ( iii ) AO , if produced , bisects BC at right angles . 13. Describe an isosceles triangle ...
Page 55
... vertical angle of a triangle is at right angles to the base , the triangle is isosceles . 6 . The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line ; and of others , that which is ...
... vertical angle of a triangle is at right angles to the base , the triangle is isosceles . 6 . The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line ; and of others , that which is ...
Page 65
... vertical angle is equal to two right angles . 3. If two straight lines are perpendicular to two other straight lines , each to each , the acute angle between the first pair is equal ... angle . 7. The angle contained BOOK I. PROP . 32 . 65.
... vertical angle is equal to two right angles . 3. If two straight lines are perpendicular to two other straight lines , each to each , the acute angle between the first pair is equal ... angle . 7. The angle contained BOOK I. PROP . 32 . 65.
Page 71
... angle ACD is obtuse . 5. Shew that in any regular polygon of n sides each angle contains right angles . 2 ( n - 2 ) n 6. The angle contained by the bisectors of the angles at the base of any triangle is equal to the vertical angle ...
... angle ACD is obtuse . 5. Shew that in any regular polygon of n sides each angle contains right angles . 2 ( n - 2 ) n 6. The angle contained by the bisectors of the angles at the base of any triangle is equal to the vertical angle ...
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Common terms and phrases
ABCD AC is equal adjacent angles Algebra angle BAC angle equal base BC bisected bisectors centre chord circumference circumscribed circle concyclic Constr Describe a circle diagonal diameter divided equal angles equiangular Euclid Euclid's exterior angle find the locus given circle given point given straight line given triangle greater Hence hypotenuse inscribed circle isosceles triangle Let ABC line which joins magnitudes meet middle point nine-points circle opposite sides orthocentre parą parallelogram parm pass pedal triangle perp perpendiculars drawn plane XY polygon produced Proof proportional PROPOSITION PROPOSITION 13 prove quadrilateral radical axis radius rectangle contained rectilineal figure regular polygon right angles segment shew shewn side BC Similarly square straight line drawn tangent THEOREM triangle ABC twice the rect vertex vertical angle
Popular passages
Page 353 - Pythagoras' theorem states that the square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
Page 340 - 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.
Page 65 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
Page 162 - AB into two parts, so that the rectangle contained by the whole line and one of the parts, shall be equal to the square on the other part.
Page 326 - From this it is manifest that prisms upon triangular bases, of the same altitude, are to one another as their bases. Let the...
Page 162 - If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together •with the square...
Page 291 - The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.
Page 79 - The straight line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of it 46 INTERCEPTS BY PARALLEL LINES.
Page 18 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Page 242 - We may here notice that the perpendiculars from the vertices of a triangle to the opposite sides are concurrent ; their meet is called the orthocentre, and the triangle obtained by joining the feet of the perpendiculars is called the pedal triangle.