. The principles of projective geometry applied to the straight line and conic . abed, the chief propertiesof Harmonic Ranges and Pencils may be stated as follows: , . 2 _ 1 1 (b) ^, (c) = OA^ = OB%.,. DC /ACy (BG\^ (^) od^Kad)=\bd)^ («) 2 cot ah =« cot ac + cot ad, {b) 2 sin ca. sin cb — sin 2 co. tan cd,(c) cot oc. cot od=cot^ oa = cot^ ob,sin 2 oc /sinca \2 /sinc6\^^sni dh . ,,, sin 2 oc _ /sin ca \^ _ /sin c6 X^sin 2 ot/ Vsin daJ Vsin ^ft/ sin 2 od Vsin ci?a> The results on the left-hand side may be deduced from the fact thatAC AD__ bcbd~ ^ and those on the right-hand ei
. The principles of projective geometry applied to the straight line and conic . abed, the chief propertiesof Harmonic Ranges and Pencils may be stated as follows: , . 2 _ 1 1 (b) ^, (c) = OA^ = OB%.,. DC /ACy (BG\^ (^) od^Kad)=\bd)^ («) 2 cot ah =« cot ac + cot ad, {b) 2 sin ca. sin cb — sin 2 co. tan cd,(c) cot oc. cot od=cot^ oa = cot^ ob,sin 2 oc /sinca \2 /sinc6\^^sni dh . ,,, sin 2 oc _ /sin ca \^ _ /sin c6 X^sin 2 ot/ Vsin daJ Vsin ^ft/ sin 2 od Vsin ci?a> The results on the left-hand side may be deduced from the fact thatAC AD__ bcbd~ ^ and those on the right-hand either from sin ac sin ad sin be sin bdor from the corresponding relation on the left-hand. (a) Express all the distances in (ABCD)^ -1 in terms of distances from after simplification it is seen that AB AC^AD(a) Let *S be the vertex of the a perpendicular DBCA on a, meetingc and b in C and B (not shown in figure).Then ABCD is harmonic. Hence 2-^ = ^, + -^, „ >yj _ >s^ SA AB~ AC^ 2 cot a6 = cot ac + cot ac^. (b) From (a) since = CA + OB,2. CD ~ CA ^ CB {b) Similarly, since CA + CB_ ~ 2co = ca + cb, sin ca. sin 2 sin ca sin eft = sin 2co tan cd. Projective Forms Harmonic 27 AC cb AD AC- -CB AD- BD (c) Since • ■ AC+CB AD + ^■■ ~ = OB^ = OA^.(c) Drop a perpendicular from D on o meeting a, b, c and o in ABCOThen 0 is the middle point of AB, therefore, since ABCD is harmonic,OC .0D = 0A\ SO SO f sa SO SO _ sav 0C0D~\OA) (d) From (c) cot oc cot od=cot oa=cot - ob.^OA ^OD^ J^OC OA -ACby subtracting numerators and denominators. OD^fADy•■ OC~\ACj OD /BD\ Similarly{d) From (c) OC cot OC _ cot oa cot oa cot od cot OC _ /sin cacot 00? ^sin da \BCsin crt sin da sin orf Similarly sin 2. OC sin 2. odsin 2. oc \- /sin OC \-/ ^sin od^_ /sin ca \2Vsin c?rt/ (sin cby-sin o?6/ sin 2. od The internal and external bisectors of any angle form luith the linesthat include the angle a harmonic pen
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
