Elements of geometry and trigonometry . PROPOSITION VI. THEOREM. If from a point without a plane, a perpendicular he let fall on theplane, and from the foot of the j)erpendicular a perpendicularbe drawn to any line of the plane, and from the point of inter-section a line be drawn to the first point, this latter line will hepeiyendicular to the line of the plane. Let AP be perpendicular to theplane NM, and PD perpendicular to]^C ; then will AD be also perpen-dicular to BC. Take DB = DC. and draw PB, PC,AB, AC. eince DB=rDC, the ob-lique line PB=:PC: and with regardto the perpendicular AP, since
Elements of geometry and trigonometry . PROPOSITION VI. THEOREM. If from a point without a plane, a perpendicular he let fall on theplane, and from the foot of the j)erpendicular a perpendicularbe drawn to any line of the plane, and from the point of inter-section a line be drawn to the first point, this latter line will hepeiyendicular to the line of the plane. Let AP be perpendicular to theplane NM, and PD perpendicular to]^C ; then will AD be also perpen-dicular to BC. Take DB = DC. and draw PB, PC,AB, AC. eince DB=rDC, the ob-lique line PB=:PC: and with regardto the perpendicular AP, since PB =PC, the oblique line AB==AC ( Cor.) ; tlierefore the line AD hastwo of its points A and D equally distant from the extremitiesB and C ; therefore AD is a perpendicular to BC, at its middlepoint D (Book I. Prop. XVI. Cor.).. Ei BOOK VI. 131 Cor. It is evident likewise, that BC is perpendicular to theplane API), since BC is at once perpendicular to the twostraight lines AD, PD. Scholium. The two lines AE, BC, afford an instance of twolines which do not meet, because they are not situated in thesame |)!aiic. The shortest distance between these lines is thestraight line PL), which is at once perpendicular to the line APand to the line BC. The distance Pi) is the shortest distancebetween them, because it we join anv other two points, suchas A and B, we shall have AB>ÀD, AD>PD; thereforeAB>PI). The two lines AE, CB, thouuli not situated in the same plane,are conceived as forming a right angle with each other, becauseAE and the line drawn through one of its points parallel toBC would make with each other a right angle. In the samemanner, the line AB and the line PI), which represent any twostraifiht lines not situated in the same plane, are supposed toform with each other the same angle, which would b
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
