Elements of geometry and trigonometry . NPD, les;thelike 130 GEOMETRY. manner, if the distance PEis greater than PD or its equal PB,the obhque hne AE will evidently be greater than AB, or itsequal AD. Cor. AH the equal obliquelines, AB, AC, AD, ifec. termi-nate inthe circumference BCD,described from P the foot of the -^j)erpendicular as a centre ;therefore a point A being givenout of a plane, the point P atwhich the perpendicular let fallfrom A would meet that plane,may be found by marking uponthat plane three points B, C, D, equally distant from the pomt A,and then finding the centre of the c
Elements of geometry and trigonometry . NPD, les;thelike 130 GEOMETRY. manner, if the distance PEis greater than PD or its equal PB,the obhque hne AE will evidently be greater than AB, or itsequal AD. Cor. AH the equal obliquelines, AB, AC, AD, ifec. termi-nate inthe circumference BCD,described from P the foot of the -^j)erpendicular as a centre ;therefore a point A being givenout of a plane, the point P atwhich the perpendicular let fallfrom A would meet that plane,may be found by marking uponthat plane three points B, C, D, equally distant from the pomt A,and then finding the centre of the circle which passes throughthese points ; this centre will be P, the point sought. Scholium. The angle ABP is called the inclination of theoblique line AB to the plane MN ; which inclination is evidentlyequal with respect to all such lines AB, AC, AD, as are equallydistant from the perpendicular ; for all the triangles ABP, ACP,ADP, &c. are equal to each 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.).
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