een lines in the same plane drawn from the same points to any other circumference but MN.
Draw a line DB outside of the figure and divide it so that D:B=MH:MK. But MH is greater than MK since the reflection of the cone is over the greater angle (for it subtends the greater angle of the triangle KMH). Therefore D is greater than B. Then add to B a line Z such that B+Z =D:B. Then make another line having the same ratio to B as KH has to Z, and join MI.
Then I is the pole of the circle on which the lines from K fall. For the ratio of D to IM is the same as that of Z to KH and of B to KI. If not, let D be in the same ratio to a line indifferently lesser or greater than IM,Buty Jordan Retro 4, and let this line be IP. Then HK and KI and IP will have the same ratios to one another as Z, B, and D. But the ratios between Z, B, and D were such that Z+B=D: B. Therefore IH:IP=IP:IK. Now, if the points K, H be joined with the point P by the lines HP, KP,Fotballdrakter 17/18, these lines will be to one another as IH is to IP, for the sides of the triangles HIP, KPI about the angle I are homologous. Therefore, HP too will be to KP as HI is to IP. But this is also the ratio of MH to MK,Parajumpers Damen Harraseeket Jacken Schweiz, for the ratio both of HI to IP and of MH to MK is the same as that of D to B. Therefore, from the points H,Belstaff Icon Bunda, K there will have been drawn lines with the same ratio to one another,Arctic Parka, not only to the circumference MN but to another point as well, which is impossible. Since then D cannot bear that ratio to any line either lesser or greater than IM (the proof being in either case the same), it follows that it must stand in that ratio to MI itself. Therefore as MI is to IK so IH will be to MI and finally MH to MK.
If,Leicester City Drakt, then, a circle be described with I as pole at the distance MI it will touch all the angles which the lines from H and K make by their reflection. If not,Schalke 04 Drakt, it can be shown, as before, that lines drawn to different points in the semicircle will have the same ratio to one another, which was impossible. If, then, the semicircle A be revolved about the diameter HKI, the lines reflected from the points H,Dame Moncler Downy Trimmed, K at the point M will have the same ratio,Menn Moncler Ghislain, and will make the angle KMH equal, in every plane. Further,Classic Tall Czyścibut 5815, the angle which HM and MI make with HI will always be the same. So there are a number of triangles on HI and KI equal to the triangles HMI and KMI. Their perpendiculars will fall on HI at the same point and will be equal. Let O be the point on which they fall. Then O is the centre of the circle,Air Jordan Retro 7 Męskie, half of which,Canada Goose Whistler Parka, MN, is cut off by the horizon. (See diagram.)
Next let the horizon be ABG but let H have risen above the horizon. Let the axis now be HI. The proof will be the same for the rest as before, but the pole I of the circle will be below the horizon AG since the point H has risen above the horizon. But the pole, and the centre of the circle, and the centre of that circle (namely HI) which now determines the position of the sun are on the same line. But since KH lies above the diameter AG, the centre will be at O on the line KI below the plane of the circle AG determined the positio
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