Transactions of the Royal Society of Edinburgh . oin B and D, the given point, in the linewhofe fegment is to be intercepted by BF. Let BD meet thecircle in K; join AK, and take AM equal to BD. Through thepoints D, M, K defcribe a circle cutting DE in L, and AK inM. Join LM, and from the point A (by prop, 12.) draw aftraight Hne to meet CE in G, and LM in N, fo that the rec-tangle 134 GEOMETRICAL PORISMS, tangle MN, CG may be equal to that which is to be containedby CG, DH. Let AN meet the circle in F; join BF meetingDEinH. The angle HDB is equal to LMK or AMN, and the angleDBH is equal to MAN
Transactions of the Royal Society of Edinburgh . oin B and D, the given point, in the linewhofe fegment is to be intercepted by BF. Let BD meet thecircle in K; join AK, and take AM equal to BD. Through thepoints D, M, K defcribe a circle cutting DE in L, and AK inM. Join LM, and from the point A (by prop, 12.) draw aftraight Hne to meet CE in G, and LM in N, fo that the rec-tangle 134 GEOMETRICAL PORISMS, tangle MN, CG may be equal to that which is to be containedby CG, DH. Let AN meet the circle in F; join BF meetingDEinH. The angle HDB is equal to LMK or AMN, and the angleDBH is equal to MAN -, now BD »is equal to AM ; thereforethe triangles BDH, AMN are in all refpeds equal, and DH isequal to MN. Therefore the redangle DH, CG is equal toMN, CG, that is, (by conflrudtion), to the given fpace as requi-red. It is eafy to fee, how, in like manner, by drawing AGN, fothat CG may be to MN in a given ratio, (prop, ii.), thelines BF, AF fhall cut off fegments DH, CG, having to eachother a given ratio. V. PL ATT. I 77/«v/,V, /,(
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