21. If two isosceles triangles be of equal altitudes, and the side of one be equal to the side of the other, their bases shall be equal. 22. If a straight line be drawn to touch a circle and parallel to a chord, the point of contact will be the middle point of the arc cut off by that chord. 23. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet to the outer circunference, is bisected by the interior one. 24. If two opposite angles of a quadrilateral figure be together equal to two right angles, a circle may be described about it. 25. Of two circular segments upon the base, the larger is that which contains the smaller angle. 26. If two circles intersect, the common chord produced bisects the common tangent. 27. Four circles are described so that each one may touch internally three of the sides of a quadrilateral; show that a circle may be described so as to pass through the four centres of the above circles. 28. If two circles cut each other, the straight line joining their centres will bisect their common chord at right angles. 29. The perpendiculars let fall from the three angles of any triangle upon the opposite sides, intersect each other in the same point. 30. If a straight line touch the interior of two concentric circles and be placed in the exterior one, it will be bisected in the point of contact. 31. If from any two points in the circumference of the greater of two given concentric circles, two straight lines be drawn so as to touch the less circle, they shall be equal to one another. 32. If a quadrilateral rectilineal figure be described about a circle, the angles subtended, at the centre of the circle, by any two opposite sides of the figure, are, together, equal to two right angles. 33. If an arc of a circle be divided into three equal parts by three straight lines drawn from one extremity of the arc, the angle contained by two of the straight lines is bisected by the third. 34. If an equilateral triangle be inscribed in a circle, the square on a side thereof is equal to three times the square described upon the radius. 35. Given the vertical angle, the base, and the altitude of a triangle, to construct it. 36. Given the vertical angle, the base, and the sum of the sides of a triangle, to construct it. 37. Describe a circle which shall touch a given circle, and also touch a given line in a given point. 38. If on the three sides of any triangle, equilateral triangles be described; straight lines joining the centres of the circles described about these three triangles, will form an equilateral triangle. 39. ABD, ACE are two straight lines touching a circle at B and C, and if DE be joined, DE is equal to BD and CE together; show that DE touches the circle. 40. The greatest rectangle which can be inscribed in a circle is a square. QUESTIONS AND EXERCISES ON BOOK IV. DEFINITIONS. IV. 1. When is a rectilineal figure said to be inscribed in another rectilineal figure? II. When is a rectilineal figure said to be described about another rectilineal figure? III. When is a rectilineal figure said to be inscribed in a circle? When is a rectilineal figure said to be described about a circle? v. When is a circle said to be inscribed in a rectilineal figure? VI. When is a circle said to be described about a rectilineal figure? VII. When is a straight line said to be placed in a circle? PROPOSITIONS AND COROLLARIES OF BOOK IV. Prop. 1. In a given circle, to place a straight line equal to a given straight line not greater than the diameter of a circle. Prop. 2. In a given circle, to inscribe a triangle equiangular to a given triangle. Prop. 3. About a given circle, to describe a triangle equiangular to a given triangle. Prop. 4. To inscribe a circle in a given triangle. Prop. 5. To describe a circle about a given triangle. Cor. When the centre of a circle falls within the triangle, each of its angles is less than a right angle; but when the centre is in one of the sides of the triangle, the angle opposite to this side is a right angle; and if the centre falls without the triangle, the angle opposite to the side beyond which it is, is greater than a right angle. And conversely if the given triangle be acuteangled the centre of the circle falls within it; if it be a right-angled triangle the centre is in the side opposite to the right angle; and if it be an obtuse-angled triangle, the centre falls without the triangle, beyond the side opposite to the obtuse angle. Prop. 6. To inscribe a square in a given circle. Prop. 10. To describe an isosceles triangle, having each of the angles at the base double of the third angle. Prop. 11. To inscribe an equilateral and equiangular pentagon in a given circle. Prop. 12. To describe an equilateral and equiangular pentagon about a given circle. Prop. 13. To inscribe a circle in a given equilateral and equiangular pentagon. Prop. 14. To describe a circle about a given equilateral and equiangular pentagon. Prop. 15. To inscribe an equilateral and equiangular hexagon in a given circle. Cor. The side of the hexagon is equal to the straight line from the centre, that is, to the semi-diameter of the circle. Prop. 16. To inscribe an equilateral and equiangular quindecagon in a given circle. GEOMETRICAL EXERCISES ON BOOK IV. 1. In a given circle, to place a straight line equal and parallel to a straight line given in position, and not greater than a diameter. 2. In a given circle place a line of given length, which shall pass through a given point. 3. An equilateral triangle is inscribed in a circle, and through the angular points tangents are drawn; show that they will form an equilateral triangle, whose area is four times the former. 4. In figure Prop. IV. 4, show that the straight line DA will bisect the angle at A. 5. In figure Prop Iv. 5, show that the perpendicular from F on BC will bisect BC. 6. In the figure Prop. Iv. 10, show that the angle A at the vertex of the triangle ABD is one-fifth of two right angles, and each of the angles at the base two-fifths of two right angles. 7. Divide a right angle into five equal parts. 8. In the figure Prop. IV. 10, show that AC is the side of a regular decagon inscribed in the larger circle. 9. In figure Prop. IV. 10, show that the angle ACD is equal to three times the angle at the vertex of the triangle. 10. On a given line to describe an equilateral and equiangular pentagon. 11. Given a regular pentagon; describe a triangle of the same area and altitude. 12. Describe an equilateral and equiangular octagon in a circle. 13. If two circles be described, one without and the other within a rightangled triangle, the sum of their diameters is equal to the sum of the sides containing the right angle. 14. Inscribe a square in a given right-angled isosceles triangle. 15. The centres of the inscribed and circumscribed circles of an equilateral triangle coincide, and the diameter of one is double that of the other. 16. The lines joining the alternate angles, or the intersections of the alternate sides of a regular pentagon, will form another regular pentagon. 17. Inscribe a circle in a given rhombus. 18. ABCDE is a regular pentagon; join AC and BE, and let BE meet AC in F; show that AC is equal to the sum of AB and BF. 19. The square inscribed in a circle is equal to half the square described about the same circle. 20. The centre of the circle which touches the two semicircles described on the sides of a right-angled triangle is the middle point of the hypotenuse. 21. A regular octagon inscribed in a circle is equal to the rectangle under the sides of the inscribed and circumscribing squares. 22. To inscribe a circle in a given quadrant. 23. The square on the side of a pentagon inscribed in a circle, is equal to the sum of the squares on the sides of a hexagon and decagon, inscribed in the same circle. 24. Describe a circle which shall pass through one angle, and touch two sides of a given square. 25. If DE be drawn parallel to the base BC of a triangle ABC, show that the circles described about the triangles ABC and ADE have a common tangent. 26. The angle ACB of any triangle is bisected, and the base AB is bisected at right angles, by straight lines which intersect at D; show that the angles ACB, ADB are together equal to two right angles. 27. If ABCDEF is a regular hexagon, and AC, BD, CE, DF, EA, FB, be joined, another hexagon will be formed whose area is one-third of that of the former. 28. If any number of parallelograms be inscribed in a given parallelogram, the diameters of all the figures shall cut one another in the same point. 29. If ABCDE be any pentagon inscribed in a circle, and AC, BD, CE, DA. EB, be joined, then are the angles ABE, BCA, CDB, DEC, EAD, together equal to two right angles. 30. If in any circle the side of an inscribed hexagon be produced till it becomes equal to the side of an inscribed square. a tangent drawn from the extremity, without the circle, shall be equal to the side of an inscribed octagon. LONDON PRINTED BY SPOTTISWOODE AND CO. NEW-STREET SQUARE GENERAL LISTS OF SCHOOL-BOOKS PUBLISHED BY Messrs. LONGMAN, GREEN, and CO. English Reading-Lesson Books. M'Leod's First Reading-Book, 18mo. 3d.; or as Reading-Lessons, in 30 Sheets, 3s. Unwin's Infant-School Reader, fcp. 8vo... Laurie's Graduated Series of Reading-Lesson Books, fep. 8vo. 6d. 9d. 6d. .10s. Book I. 1s.; Book II. 1s. 6d.; Book III. 2s.; Book IV. 2s. 6d.; Book V. 3s. Laurie's Shilling Entertaining Library: Robinson Crusoe; Gulliver's Travels; Christmas Tales: Sandford and Merton ... each sewed, 9d.; cioth, 1s. 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