Book IV. DE is equal to DF. for the fame reason, DG is equal to DF; there fore the three straight lines DE, DF, DG are equal to one another, and the circle defcribed from the center D, at the distance of any of F, G are right angles, and the E straight line which is drawn from the extremity of a diameter at right d. 16. 3. angles to it, touches the circle. See N. a. 10. I. b. II. I. & B therefore the straight lines AB, BC, A F C CA do each of them touch the circle, and the circle EFG is infcribed in the triangle ABC. Which was to be done. PROP. V. PROB. To defcribe a circle about a given triangle. Let the given triangle be ABC; it is required to defcribe a circle about ABC. Bifecta AB, AC in the points D, E, and from these points draw DF, EF at right angles to AB, AC; DF, EF produced C. 4. I. 1 meet one another. for if they do not meet they are parallel, wherefore AB, AC which are at right angles to them are parallel; which is abfurd. let them meet in F, and join FA; alfo, if the point F be not in BC; join BF, CF. then because AD is equal to DB, and DF common, and at right angles to AB, the base AF is equal to the base FB. in like manner it may be fhewn that CF is equal to FA; and therefore BF is equal to FC; and FA, FB, c FC are therefore equal to one another. wherefore the circle de- Book IV. fcribed from the center F, at the distance of one of them, fhall pass thro' the extremities of the other two; and be defcribed about the triangle ABC. Which was to be done. COR. And it is manifeft that when the center of the circle falls within the triangle, each of its angles is lefs than a right angle, each of them being in a fegment greater than a femicircle. but when the center is in one of the fides of the triangle, the angle opposite to this fide, being in a femicircle, is a right angle. and if the center falls without the triangle, the angle oppofite to the fide beyond which it is, being in a fegment lefs than a femicircle, is greater than a right angle. Wherefore, if the given triangle be acute angled, the center of the circle falls within it; if it be a right angled triangle, the center is in the fide oppofite to the right angle; and if it be an obtufe angled triangle, the center falls without the triangle, beyond the fide oppofite to the obtuse angle. PROP. VI. PROB. To infcribe a fquare in a given circle. Let ABCD be the given circle; it is required to infcribe a fquare in ABCD. Draw the diameters AC, BD at right angles to one another; and join AB, BC, CD, DA. because BE is equal to ED, for E is the center, and that EA is common, and at right angles to BD; the bafe BA is A E a. 4. I. the angle BAD is a right angle. for the fame reafon each of the b. 31. 3. angles ABC, BCD, CDA is a right angle. therefore the quadrilateral figure ABCD is rectangular. and it has been shewn to be equilateral, therefore it is a fquare; and it is inferibed in the circle ABCD. Which was to be done. Book IV. PROP. VII. PROB. To defcribe a square about a given circle. Let ABCD be the given circle; it is required to describe a fquare about it. Draw two diameters AC, BD of the circle ABCD, at right 2. 17. 3. angles to one another, and thro' the points A, B, C, D draw a FG, GH, HK, KF touching the circle. and because FG touches the circle ABCD, and EA is drawn from the center E to the point of b. 18. 3. contact A, the angles at A are right angles. for the fame reafon, the angles at the points B, C, D are right angles. and because the c. 28. I. d. 34. I. 4. 10. I. C angle AEB is a right angle, as likewise G A E and H F K is EBG, GH is parallel to AC. for the PROP. VIII. PROB. To infcribe a circle in a given square. Let ABCD be the given square; it is required to inscribe a circle in ABCD. Bifect each of the fides AB, AD, in the points F, E, and thro’E b. 31. 1. draw b EH parallel to AB or DC, and thro' F draw FK parallel to A C. 34. I. AD or BC. therefore each of the figures AK, KB, AH, HD, AG, Book IV. GC, BG, GD is a parallelogram, and their oppofite fides are equal. and because AD is equal to AB, and that AE is the half of AD, and AF the half of AB; AE is equal to AF. wherefore the fides oppofite to these are equal, viz. FG to GE. in the fame manner it may be demonstrated that GH, GK are each of them equal to FG or GE. therefore the four ftraight lines' GE, GF, GH, GK are equal to one another; and the circle defcribed from the center G, at the distance of one of them shall pass thro' the extremities of the other or F B d E Ꭰ G K C H three, and touch the straight lines AB, BC, CD, DA; because the angles at the points E, F, H, K are right angles, and that the d. 29. 1. straight line which is drawn from the extremity of a diameter, at right angles to it, touches the circle . therefore each of the e. 16. 3. straight lines AB, BC, CD, DA touches the circle, which therefore is inscribed in the square ABCD. Which was to be done. PROP. IX. PROB. To defcribe a circle about a given square. Let ABCD be the given square; it is required to describe a circle about it. Join AC, BD cutting one another in E. and because DA is equal to AB, and AC common to the triangles DAC, BAC, the two fides DA, AC are equal to the two BA, AC; and the base DC is equal to the base BC; wherefore the angle DAC is equal a to the angle BAC, and the angle DAB is bifected by the straight line AC. in the fame manner it may be demonstrated that the angles ABC, BCD, CDA are feverally bifected by B the straight lines BD, AC. therefore be a. 8. z. E cause the angle DAB is equal to the angle ABC, and that the angle EAB is the half of DAB, and EBA the half of ABC; the angle EAB is equal to the angle EBA; wherefore the fide EA is equal to the b. 6. za Lide EB. in the fame manner it may be demonftrated that the straight 5 Book IV. lines EC, ED are each of them equal to EA or EB. therefore the four straight lines EA, EB, EC, ED are equal to one another; and the circle described from the center E, at the distance of one of them, fhall pass thro' the extremities of the other three, and be described about the fquare ABCD. Which was to be done. a. II. 2, b. I. 4. c. 5.4. d. 37. 3. e. 32. 3. f. 32. I. O describe an isofceles triangle, having each of the T° b Take any straight line AB, and divide a it in the point C, fo that the rectangle AB, BC be equal to the fquare of CA; and from the center A, at the distance AB defcribe the circle BDE, in which place the straight line BD equal to AC, which is not greater than the diameter of the circle BDE; join DA, DC, and about the triangle ADC describe the circle ACD. the triangle ABD is fuch as is required, that is, each of the angles ABD, ADB is double of the angle BAD. E A Because the rectangle AB, BC is equal to the fquare of AC, and -that AC is equal to BD, the rectangle AB, BC is equal to the fquare of BD. and because from the point B without the circle ACD two ftraight lines BCA, BD are drawn to the circumference, one of which cuts, and the other meets the circle, and that the rectangle AB, BC contained by the whole of the cutting line, and the part of it without the circle, is equal to the fquare of BD which meets it; the ftraight line BD touches the circle ACD. and because BD touches the circle, and DC is drawn from the point of contact D, the angle BDC is equal to the angle DAC in the alternate fegment of the circle; to each of these add the angle CDA, therefore the whole angle BDA is equal to the two angles CDA, DAC. but the exterior angle BCD is equal f to the angles CDA, DAC; therefore alfo BDA is equal to BCD, but BDA is e B D |