but the angle ABE has been proved equal to BAH: therefore the angle BEA is equal to the angle BAH: and ABE is common to the two triangles ABE, ABH; therefore the remaining angle BAE is equal to the remaining angle AHB; and consequently the triangles ABE, ABH are equiangular; therefore EB is to BA, as AB to BH: but BA is equal to EH, therefore EB is to EH, as EH is to BH, but BE is greater than EH; therefore EH is greater than HB; therefore BE has been cut in extreme and mean ratio in H. Similarly, it may be shewn, that AC has also been cut in extreme and mean ratio in H, and that the greater segment of it CH is equal to the side of the pentagon. PROPOSITION IV. PROBLEM. Divide a given arc of a circle into two parts which shall have their chords in a given ratio. Analysis. Let A, B be the two given points in the circumference of the circle, and Cthe point required to be found, such that when the chords AC and BC are joined, the lines AC and BC shall have to one another the ratio of E to F. Draw CD touching the circle in C; join AB and produce it to meet CD in D. Since the angle BAC is equal to the angle BCD, (III. 32.) and the angle CDB is common to the two triangles DBC, DAC; therefore the third angle CBD in one, is equal to the third angle DCA in the other, and the triangles are similar, therefore AD is to DC, as DC is to DB; (VI. 4.) hence also the square on AD is to the square on DC, as AD is to BD. (VI. 20. Cor.) But AD is to AC, as DC is to CB, (vI. 4.) and AD is to DC, as AC to CB, (v. 16.) also the square on AD is to the square on DC, as the square on AC is to the square on CB; but the square on AD is to the square on DC, as AD is to DB: wherefore the square on AC is to the square on CB, as AD is to BD; but AC is to CB, as E is to F, (constr.) therefore AD is to DB as the square on E is to the square on F. Hence the ratio of AD to DB is given, and AB is given in magnitude, because the pointsA, B in the circumference of the circle are given. Wherefore also the ratio of AD to AB is given, and also the mag nitude of AD. Synthesis. Join AB and produce it to D, so that AD shall be to BD, as the square on E to the square on F. From D draw DC to touch the circle in C, and join CB, CA. Since AD is to DB, as the square on E is to the square on F, (constr.) and AD is to DB, as the square on AC is to the square on BC; therefore the square on AC is to the square on BC, as the square on E is to the square on F, and AC is to BC, as E is to F. PROPOSITION V. PROBLEM. A, B, C are given points. It is required to draw through any other point in the same plane with A, B, and C, a straight line, such that the sum of its distances from two of the given points, may be equal to its distance from the third. Analysis. Suppose F the point required, such that the line XFH being drawn through any other point X, and AD, BE, CH perpendiculars on XFH, the sum of BE and CH is equal to AD. Join AB, BC, CA, then ABC is a triangle. Draw AG to bisect the base BC in G, and draw GK perpendicular to EF. Then since BC is bisected in G, the sum of the perpendiculars CH, BE is double of GK; but AD is double of GK, therefore AF is double of GF; and consequently, GF is one-third of AG the line drawn from the vertex of the triangle to the bisection of the base. But AG is a line given in magnitude and position, Synthesis. Join AB, AC, BC, and bisect the base BC of the triangle ABC in G; join AG and take GF equal to one-third of GA; the line drawn through X and F will be the line required. It is also obvious, that while the relative position of the points A, B, C, remains the same, the point Fremains the same, wherever the point X may be. The point X may therefore coincide with the point F, and when this is the case, the position of the line FX is left undetermined. Hence the following porism. A triangle being given in position, a point in it may be found, such, that any straight line whatever being drawn through that point, the perpendiculars drawn to this straight line from the two angles of the triangle, which are on one side of it, will be together equal to the perpendicular that is drawn to the same line from the angle on the other side of it. I. 6. TRIANGLES and parallelograms of unequal altitudes are to each other in the ratio compounded of the ratios of their bases and altitudes. 7. If ACB, ADB be two triangles upon the same base AB, and between the same parallels, and if through the point in which two of the sides (or two of the sides produced) intersect two straight lines be drawn parallel to the other two sides so as to meet the base AB (or AB produced) in points E and F. Prove that AE= BF. 8. In the base AC of a triangle ABC take any point D; bisect AD, DC, AB, BC, in E, F, G, H respectively: shew that EG is equal to HF. 9. Construct an isosceles triangle equal to a given scalene triangle and having an equal vertical angle with it. 10. If, in similar triangles, from any two equal angles to the opposite sides, two straight lines be drawn making equal angles with the homologous sides, these straight lines will have the same ratio as the sides on which they fall, and will also divide those sides proportionally. 11. Any three lines being drawn making equal angles with the three sides of any triangle towards the same parts, and meeting one another, will form a triangle similar to the original triangle. 12. BD, CD are perpendicular to the sides AB, AC of a triangle ABC, and CE is drawn perpendicular to AD, meeting AB in E: shew that the triangles ABC, ACE are similar. 13. In any triangle, if a perpendicular be let fall upon the base from the vertical angle, the base will be to the sum of the sides, as the difference of the sides to the difference or sum of the segments of the base made by the perpendicular, according as it falls within or without the triangle. 14. If triangles AEF, ABC have a common angle A, triangle ABC: triangle AEF:: AB.AC: AE.AF. 15. If one side of a triangle be produced, and the other shortened by equal quantities, the line joining the points of section will be divided by the base in the inverse ratio of the sides. II. 16. Find two arithmetic means between two given straight lines. 17. To divide a given line in harmonical proportion. 18. To find, by a geometrical construction, an arithmetic, geometric, and harmonic mean between two given lines. 19. Prove geometrically, that an arithmetic mean between two quantities, is greater than a geometric mean. Also having given the sum of two lines, and the excess of their arithmetic above their geometric mean, find by a construction the lines themselves. 20. If through the point of bisection of the base of a triangle any line be drawn, intersecting one side of the triangle, the other produced, and a line drawn parallel to the base from the vertex, this line shall be cut harmonically. 21. If a given straight line AB be divided into any two parts in the point C, it is required to produce it, so that the whole line produced may be harmonically divided in Cand B. 22. If from a point without a circle there be drawn three straight lines, two of which touch the circle, and the other cuts it, the line which cuts the circle will be divided harmonically by the convex circumference, and the chord which joins the points of contact. III. 23. Shew geometrically that the diagonal and side of a square are incommensurable. 24. If a straight line be divided in two given points, determine a third point, such that its distances from the extremities, may be proportional to its distances from the given points. 25. Determine two straight lines, such that the sum of their squares may equal a given square, and their rectangle equal a given rectangle. 26. Draw a straight line such that the perpendiculars let fall from any point in it on two given lines may be in a given ratio. 27. If diverging lines cut a straight line, so that the whole is to one extreme, as the other extreme is to the middle part, they will intersect every other intercepted line in the same ratio. 28. It is required to cut off a part of a given line so that the part cut off may be a mean proportional between the remainder and another given line. 29. It is required to divide a given finite straight line into two parts, the squares of which shall have a given ratio to each other. IV. 30. From the vertex of a triangle to the base, to draw a straight line which shall be an arithmetic mean between the sides containing the vertical angle. 31. From the obtuse angle of a triangle, it is required to draw a line to the base, which shall be a mean proportional between the segments of the base. How many answers does this question admit of? 32. To draw a line from the vertex of a triangle to the base, which shall be a mean proportional between the whole base and one segment. 33. If the perpendicular in a right-angled triangle divide the hypotenuse in extreme and mean ratio, the less side is equal to the alternate segment. 34. From the vertex of any triangle ABC, draw a straight line meeting the base produced in D, so that the rectangle DB. DC= AD2. 35. To find a point P in the base BC of a triangle produced, so that PD being drawn parallel to AC, and meeting A B produced to D, AC: CP:: CP: PD. 36. If the triangle ABC has the angle at C a right angle, and from Ca perpendicular be dropped on the opposite side intersecting it in D, then AD: DB:: AC2; CB2. 37. In any right-angled triangle, one side is to the other, as the excess of the hypotenuse above the second, to the line cut off from the first between the right angle and the line bisecting the opposite angle. 38. If on the two sides of a right-angled triangle squares be described, the lines joining the acute angles of the triangle and the opposite angles of the squares, will cut off equal segments from the sides; and each of these equal segments will be a mean proportional between the remaining segments. 39. In any right-angled triangle ABC, (whose hypotenuse is AB) bisect the angle A by AD meeting CB in D, and prove that 2AC: AC - CD2 :: BC: CD. 40. On two given straight lines similar triangles are described. Required to find a third, on which, if a triangle similar to them be described, its area shall equal the difference of their areas. 41. In the triangle ABC, AC-2.BC. If CD, CE respectively bisect the angle C, and the exterior angle formed by producing 4C; prove that the triangles CBD, ACD, ABC, CDE, have their areas as 1, 2, 3, 4. V. 42. It is required to bisect any triangle (1) by a line drawn parallel, (2) by a line drawn perpendicular, to the base. 43. To divide a given triangle into two parts, having a given ratio to one another, by a straight line drawn parallel to one of its sides. 44. Find three points in the sides of a triangle, such that, they being joined, the triangle shall be divided into four equal triangles. 45. From a given point in the side of a triangle, to draw lines to the sides which shall divide the triangle into any number of equal parts. 46. Any two triangles being given, to draw a straight line parallel to a side of the greater, which shall cut off a triangle equal to the less. VI. 47. The rectangle contained by two lines is a mean proportional between their squares. 48. Describe a rectangular parallelogram which shall be equal to a given square, and have its sides in a given ratio. 49. If from any two points within or without a parallelogram, straight lines be drawn perpendicular to each of two adjacent sides and intersecting each other, they form a parallelogram similar to the former. 50. It is required to cut off from a rectangle a similar rectangle which shall be any required part of it. |