5. If AB be produced both ways to meet the two circles again at D and E, prove that the straight line DE is equal to the sum of the three sides of the triangle ABC. 6. Show how to find a straight line equal to the sum of the three sides of any triangle. Show how to find a straight line which shall be: 7. Twice as great as a given straight line. PROPOSITION 2. PROBLEM. From a given point to draw a straight line equal to a given straight line. F H G E Let A be the given point, and BC the given straight line : it is required to draw from A a straight line Join AB, Post. 1 and on it describe the equilateral ▲ DBA. I. 1 Post. 3 Post. 2 With centre D, and radius DE, describe the OEGH; Post. 3 and produce DA to meet the ∞ EGH in G. DB = Post. 2 Because DE = and = BC. remainder AG. I. Def. 23 I. Ax. 3 BC, being radii of O CEF; I. Def. 16 I. Ax. 1 1. If the radius of the large circle be double the radius of the small circle, where will the given point be? 2. AB is a given straight line; show how to draw from A any number of straight lines equal to AB. 3. AB is a given straight line; show how to draw from B any number of straight lines equal to AB. 4. AB is a given straight line; show how to draw through A any number of straight lines double of AB. 5. AB is a given straight line; show how to draw through B any number of straight lines double of AB. 6. On a given straight line as base, describe an isosceles triangle each of whose sides shall be equal to a given straight line. May the second given straight line be of any size? If not, how large or how small may it be? Give the construction and proof of the proposition 7. When the equilateral triangle ABD is described on that side of AB opposite to the one given in the text. 8. When the equilateral triangle ABD is described on the same side of AB as in the text, but when its sides are produced through the vertex and not beyond the base. 9. When the equilateral triangle ABD is described on that side of AB opposite to the one given in the text, and when its sides are produced through the vertex. 10. When the given point A is joined to C instead of B. Make diagrams for all the cases that can arise by describing the equilateral triangle on either side of AC, and producing its sides either beyond the base or through the vertex. PROPOSITION 3. PROBLEM. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which AB is the greater: it is required to cut off from AB a part = C. = C; I. 2 From A draw the straight line AD with centre A and radius AD, describe the O DEF, Post. 3 cutting AB at E. For AE AD, being radii of O DEF. = AE shall = C. I. Def. 16 I. Ax. 1 1. Give the construction and the proof of this proposition, using the point B instead of the point A. 2. Produce the less of two given straight lines so that it may be equal to the greater. 3. If from AB (fig. 1 and fig. 2) there be cut off AD and BE, each equal to C, prove AE = BD. 4. Show how to find a straight line equal to the sum of two given straight lines. 5. Show how to find a straight line equal to the difference of two given straight lines. 6. Show that if the difference of two straight lines be added to the sum of the two straight lines, the result will be double of the greater straight line. 7. Show that if the difference of two straight lines be taken away from the sum of the two straight lines, the result will be double of the less straight line. If two sides and the contained angle of one triangle be equal to two sides and the contained angle of another triangle, the two triangles shall be equal in every respect-that is, (1) The third sides shall be equal, (2) The remaining angles of the one triangle shall be equal to the remaining angles of the other triangle, (3) The areas of the two triangles shall be equal. A A F In As ABC, DEF, let AB = DE, AC = DF, LA= LD: = it is required to prove BC = EF, LB = LE, LC LF, ▲ ABC = ▲ DEF. If A ABC be applied to A DEF, so that A falls on D, and so that AB falls on DE; = DE. Нур. Hyp. .. AC will fall on DF. And because AC = DF, Hyp. ... C will coincide with F. Now, since B coincides with E, and C with F, .. BC will coincide with EF ; F I. Def. 3 I. Ax. 9 LC and ▲ ABC will coincide with ▲ DEF; .. Δ ABC = Δ DEF. In the two As ABC, DEF, 1. If AB = DE, AC = DF, but A greater than ▲ D, where would AC fall when ABC is applied to DEF as in the proposition? 2. If AB = DE, AC = DF, but A less than D, where would AC fall? 3. If AB = DE, LA = LD, but AC greater than DF, where would fall? 4. If AB DE, LA = LD, but AC less = would C fall? than DF, where 5. Prove the proposition beginning the superposition with the point B or the point C instead of the point A. 6. If the straight line CD bisect the straight line AB perpendicularly, prove any point in CD equidistant from A and B. 7. CA and CB are two equal straight lines drawn from the point C, and CD is the bisector of ACB. Prove that any point in CD is equidistant from A and B. 8. The straight line that bisects the vertical angle of an isosceles triangle bisects the base and is perpendicular to the base. If 9. ABCD is a quadrilateral, one of whose diagonals is BD. AB = CB, and BD bisects ABC, prove that AD = CD, and that BD bisects also 4 ADC. 10. Prove that the diagonals of a square are equal. |