Elements of geometry and trigonometry . yramid has for its base the lowerbase FGH of the frustum ; its altitudelikewise is that of the frustum, becausethe vertex g lies in the plane of the up-per base fgh. This pyramid being cut off, there willremain the quadrangular pyramid g-fhllF, whose vertex is g, and base /7/HF. Pass the plane^H througli the three points f, g, H ; it will divide the quad-rangular pyramid into two triangular pyramids g-Y/H, g-fliï latter has for its base the upper base gfh of the frustum ;and for its altitude, the altitude of the frustum, because its ver-tex H lies i
Elements of geometry and trigonometry . yramid has for its base the lowerbase FGH of the frustum ; its altitudelikewise is that of the frustum, becausethe vertex g lies in the plane of the up-per base fgh. This pyramid being cut off, there willremain the quadrangular pyramid g-fhllF, whose vertex is g, and base /7/HF. Pass the plane^H througli the three points f, g, H ; it will divide the quad-rangular pyramid into two triangular pyramids g-Y/H, g-fliï latter has for its base the upper base gfh of the frustum ;and for its altitude, the altitude of the frustum, because its ver-tex H lies in the lower base. Thus we already know two ofthe three pyramids which compose the frustum. It remaiiiG to examine the third g-FJU. Now, if ^K bedrawn parallel to /F, and if we conceive a new pyramidK-F/H, having K for its vertex and F/1Ï for its base, thesetwo pyramids will have the same base F/H ; they will alsohave thp same altitude, because their vertices g and K lie intlie line ^K, parallel to F/, and consequently parallel to the. LOOK VII. 163 pîane of the base: hence these pyramids are equivalent. Butihe j)yramid K-F/H may be regarded as having its vertex inJ\ and thus its altitude \vill be the same as that of the frustum:as to its base FKH, we are now to show that this is a meanproportional between the bases FGII and fgh. Now, the tri-angles FIIK,/^/^ liave each an equal angle F=/; hence FIIK : /-//:: FKx FII : fg^fk (Book IV. Prop. XXIV.) ;but because of the parallels, FK=:4, henceFIIK :fgh : /vil : have also, FUG : FIIK : : FG : FK or the similar triangles FGII. /^/^ give FG : ./V : ; FII : fk ;hence, FGII : FIIK : : FIIK : fglr,or the base FHK is a mean proportional between the twobases FGII, fgh. Hence the frustum of a triangular pyramidis equivalent to three pyramids whose common altitudt; is thatof the frustum and whose bases arc the lower base of thefrustum, the upper base, and a mean proportional between thetwo bases. PROPOSITION XIX. THEOREM. Sim
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