Geometrical Problems Deducible from the First Six Books of Euclid, Arranged and Solved: To which is Added an Appendix Containing the Elements of Plane Trigonometry ... |
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Page vi
... pass through the same points . 16. From two given points on the same side of a line given in position , to draw two straight lines which shall contain a given angle , and be terminated in that line . 17. If from the extremities of any ...
... pass through the same points . 16. From two given points on the same side of a line given in position , to draw two straight lines which shall contain a given angle , and be terminated in that line . 17. If from the extremities of any ...
Page viii
... pass through the point of contact . 36. If two circles touch each other and also touch a straight line ; the part of ... passing through the point of contact ; the parts of the line intercepted between the circumference of this circle ...
... pass through the point of contact . 36. If two circles touch each other and also touch a straight line ; the part of ... passing through the point of contact ; the parts of the line intercepted between the circumference of this circle ...
Page ix
... passing through the point of contact a perpendicular be drawn , meeting the circumferences of the other two circles ; this diameter and the lines joining the points of intersection and contact are in continued proportion . 41. If a ...
... passing through the point of contact a perpendicular be drawn , meeting the circumferences of the other two circles ; this diameter and the lines joining the points of intersection and contact are in continued proportion . 41. If a ...
Page x
... passes through the given point , may together be equal to a given line , not greater than the diameter of the circle . 48. If from each extremity of any number of equal adjacent arcs in the circumference of a circle , lines be drawn ...
... passes through the given point , may together be equal to a given line , not greater than the diameter of the circle . 48. If from each extremity of any number of equal adjacent arcs in the circumference of a circle , lines be drawn ...
Page xi
... passes through the centre of the other , and from either point of intersection a straight line be drawn cutting both circumferences ; the part intercepted between the two circumferences will be equal to the chord drawn from the other ...
... passes through the centre of the other , and from either point of intersection a straight line be drawn cutting both circumferences ; the part intercepted between the two circumferences will be equal to the chord drawn from the other ...
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Common terms and phrases
ABCD angle ABC base bisects the angle centre chord circle ABC circles cut circles touch circumference describe a circle divided draw any line drawn parallel duplicate ratio equal angles equiangular Eucl extremities given angle given circle given in position given line given point given ratio given straight line given triangle inscribed intercepted isosceles triangle Join AE Join BD Let AB Let ABC let fall line given line joining line required lines be drawn lines drawn mean proportional opposite side parallel to BC parallelogram pendicular perpendicular be drawn point of bisection point of contact point of intersection quadrant radius rectangle contained right angles right-angled triangle segments semicircle shewn tangent touches the circle trapezium triangle ABC vertex vertical angle
Popular passages
Page 124 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight '.line which joint the points of section, shall be parallel to the remaining side of the triangle.
Page xiii - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Page 158 - Iff a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other...
Page 160 - Upon a given straight line, to describe a segment of a circle, containing an angle equal to a given angle. Let AB be the given straight line, and C the given angle ; it is required to.
Page 230 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Page 157 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Page 33 - FC ; (ax. 1.) and FA, FB, FC are equal to one another : wherefore the circle described from the centre F, at the distance of one of them, will pass through the extremities of the other two, and be described about the triangle ABC.
Page xxv - ... the squares of the diagonals, is equal to the sum of the squares of the bisected sides together with four times the square of the line joining those points of bisection.
Page 248 - If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of bisection shall be trisected by the sides.
Page 355 - The sine of an angle is equal to the sine of its supplement. The sine rule Consider fig.