GEOMETRY. 



square on EB will be equal to the given rectangle, 

 for (Prop. XXIII. i) AB BD = EB 3 . 



(4.) To divide a line into two parts, so that the 

 rectangle contained by the whole line and one of 

 the parts may be equal to the square of the other 

 part. 



Let AB be the given line ; it is F 

 required to divide it into two parts 

 in H, so that AB BH = AH 2 . 

 On AB describe the square AD. A 

 Bisect AC in E. Join EB, and 

 produqe EA to F, making EF E 

 equal to EB. Upon AF describe 

 the square FH. H is the point 

 required. For (Prop. XXI.) c 



CF FA + AE 8 = EF 2 



= AE 3 + AB 2 (Prop. XXII.) j 



/. CF FA = AB 2 . 

 Take away the common part AK, and we have 



HD = AH% 

 or AB BH = AH 2 . 



(5.) In a given circle to inscribe a triangle 

 equiangular to a given triangle. 



Let ABC be the given triangle; it is required to 

 inscribe within the given circle SDE a triangle 

 equiangular to ABC. Take any point S on the 



circumference, through it draw a tangent PQ ; at 

 S in the line PQ, make the angle QSE equal to 

 B, and PSD equal to C. Join DE. Then (Prop. 

 XV. 4) SDE is evidently the triangle required. 



(6.) About a given circle to describe a triangle 

 equiangular to a given triangle. 



Draw any tangent to the circle. At any point 

 S in this line make the angle TSH equal to the 

 angle B, and at any point T make the angle STK 



equal to C. If the lines SH, TK touch the circle, 

 the thing is done ; but if they do not, from the 

 centre O draw OF and OG perpendicular to SH 

 and TK, meeting the circle in D and E. Through 

 D and E draw tangents NM and NQ. Then 

 (Prop. IX.) NMQ is the triangle required. 



(7.) To describe a circle 

 about a given triangle. 



Bisect the sides AB and 

 AC, and through the points 

 of intersection draw perpen- 

 diculars to these sides. Let 

 these perpendiculars meet in 

 O. Then (Prop. XII.) O is 



-..A 



\ 



the centre of the circle circumscribing the 

 triangle. 



(8.) To inscribe a 

 circle within a tri- 

 angle. 



Bisect the angles B 

 and C by the lines 

 BO and CO, meeting 

 in O. Then O is the 



centre of the circle 



inscribed within the 

 triangle. 



(9.) To describe an isosceles triangle, having 

 each of the angles at the base double of the vertical 

 angle. 



Take any line AB, divide it 

 into two parts in C, so that 

 AB BC may be equal to the 

 square of AC. From A as 

 centre, with AB as radius, de- 

 scribe a circle, and in this 

 circle place the chord BD 

 equal to AC. Join AD. Then ABD is the 

 isosceles triangle required. The demonstration 

 of this is rather long and complex ; the student 

 may exercise his ingenuity in finding it out for 

 himself. To do so, he must join CD, and describe 

 a circle about the triangle ACD. The truth of it 

 being in the meantime taken for granted, it is 

 manifest that the angle at A is one-fifth of two 

 right angles, or one-tenth of four. Hence ten 

 such angles may be put round the point A, and 

 BD will be the side of a regular decagon within 

 the circle. 



(10.) To inscribe a regular 

 decagon within a circle. 



Take the radius AB, divide | 

 it in C, as in the last problem, 

 and inscribe the chords BD, 

 DE, EF, &c. each equal to AC, 

 and the thing is done. 



(u.) To inscribe a regular 

 pentagon within a circle. 



Inscribe the regular decagon, 

 and join the alternate vertices. 



(12.) To inscribe a regular 

 hexagon within the circle. 



Inscribe a chord equal to the 

 radius ; this is evidently the 

 side of a regular hexagon. 



RATIO PROPORTION. 



Definition I. The first of four magnitudes is- 

 said to have to the second the same ratio which 

 the third has to the fourth ; when, if the first be 

 divided into any number whatever of equal parts,, 

 and the third be divided into the same number of 

 equal parts, the second contains the same integral 

 number of the former parts as the fourth does of 

 the latter. 



Definition 2. Magnitudes which have the same 

 ratio are called proportionals. 



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