PROPOSITION I. PROBLEM. In a given circle to draw a chord equal to a given straight line, which is not greater than the diameter of the circle. E Let ABC be the given O, and D the given line, not greater than the diameter of the O. It is required to draw in the ○ ABC a chord=D. Draw EC, a diameter of ABC. Then if EC=D, what was required is done. But if not, EC is greater than D. From EC cut off EF=D, and with centre E and radius EF describe a O AFB, cutting ABC in A and B ; and join AE. the Thus a chord EA equal to D has been drawn in ABC. Q. E. F. Ex. Draw the diameter of a circle, which shall pass at a given distance from a given point. PROPOSITION II. PROBLEM. In a given circle to inscribe a triangle, equiangular to a given triangle. Let ABC be the given O, and DEF the given ▲. It is required to inscribe in ABC a ▲, equiangular to A DEF. Draw GAH touching the ABC at the pt. A. III. 17. I. 23. Make GAB▲ DFE, and ▲ HAC= ▲ DEF. L Join BC. Then will ▲ ABC be the required ▲. For GAH is a tangent, and AB a chord of the, ..▲ ABC is equiangular to ▲ DEF, and it is inscribed in the ABC. Q. E. F. Ex. If an equilateral triangle be inscribed in a circle, prove that the radii, drawn to the angular points, bisect the angles of the triangle. PROPOSITION III. PROBLEM. About a given circle to describe a triangle, equiangular to a given triangle. M Ꮮ B N G E H Let ABC be the given O, and DEF the given ▲. It is required to describe about the a▲ equiangular to A EDF. From O, the centre of the O, draw any radius OC. Make COA ▲ DEG, and ▲ COB= 2 DFH. = I. 23. Through A, B, C draw tangents to the O, meeting in L, M, N. Then will LMN be the ▲ required. For ML, LN, NM are tangents to the O, .. thes at A, B, C are rt. 4 s. Now 4s of quadrilateral AOCM together four rt. and of these 4 OAM and 4 OCM are rt. 4s; .. sum of 4 s COA, AMC=two rt. 4 s. But sum of 4S DEG, DEF=two rt. <s; III. 18. ≤ s.; .. sum of 4s COA, AMC=sum of 4 s DEG, DEF, Similarly, it may be shewn that LNM DFE; .. also ▲ MLN = ▲ EDF. =4 I. 32. Thus a A, equiangular to ▲ DEF, is described about the . Q. E. F. in the ▲ ABC. It is required to inscribe a Bisects ABC, ACB by the st. lines BO, CO, meeting in 0. From O draw OD, OE, OF, 1s to AB, BC, CA. Then, in s EBO, DBO, I. 9. I. 12. : 4 EBO= ▲ DBO, and ▲ BEO= ▲ BDO, and OB is common, .. OE=OD. Similarly it may be shewn that OE=OF. I. 26. If then a be described, with centre 0, and radius OD, this will pass through the pts. D, E, F ; and the s at D, E and F are rt. 4s, and thus a DEF may be inscribed in the ▲ ABC. III. 16. Q. E. F. Ex. 1. Shew that, if OA be drawn, it will bisect the angle BAC. Ex. 2. If a circle be inscribed in a right-angled triangle, the difference between the hypotenuse and the sum of the other sides is equal to the diameter of the circle. Ex. 3. Shew that, in an equilateral triangle, the centre of the inscribed circle is equidistant from the three angular points. Ex. 4. Describe a circle, touching one side of a triangle and the other two produced. (NOTE. This is called an escribed circle.) NOTE. Euclid's fifth Proposition of this Book has been already given on page 135. PROPOSITION VI. PROBLEM. To inscribe a square in a given circle. B D Let ABCD be the given O. It is required to inscribe a square in the . Through 0, the centre, draw the diameters AC, BD, 1 to each other. Join AB, BC, CD, DA. Then the s at O are all equal, being rt. 4 s, .. the arcs AB, BC, CD, DA are all equal, and.. the chords AB, BC, CD, DA are all equal; and ABC, being the in a semicircle, is a rt. 4. I. Post. 4. III. 26. III. 29. III. 31. So also the 4s BCD, CDA, DAB are rt. 4s; and it is inscribed in the O as was required. Q. E. F. |