CHAMBERS'S INFORMATION FOR THE PEOPLE. 



from the external point O, and from the same 

 point let any other line OB 

 be drawn. To prove 



(i.) That OA is less than 

 OB. Let OA be produced 

 to O', so that O'A may be 

 equal to OA. Join O'B. 

 Then OB (Prop. VII. i, p. 

 612) is equal to O'B, and 

 the two sides OB + O'B 

 are together greater than 

 OO' ; therefore the half of OB + O'B that is, 

 OB is greater than the half of OO' that is, OA. 



(2.) That AB being equal to AC, OB shall be 

 equal to OC. Because in the two triangles OAB, 

 OAC, we have two sides in the one equal to two 

 sides in the other, and the contained angles equal ; 

 therefore the third side OB shall be equal to the 

 third side OC. 



Conversely, if in the two right-angled triangles 

 OAB, OAC we have OB equal to OC, then (Prop. 

 VII. 5, p. 612) shall AB be equal to AC. 



Cor. It follows that a line OD meeting the 

 plane at D, so that AD is 

 greater than AC ; then 

 shall OD be greater than 

 OC. Cut off from AD the 

 part AB equal to AC. Then 

 OB is equal to OC. But 

 OA, OB, OD being all in 

 one plane, and the angle 

 OAD a right angle, OD is 

 greater than OB that is, OD is greater than OC. 



Scholium. The distance of a point from a plane 

 is estimated by the length of the perpendicular 

 drawn from the point to the plane. 



Definition. A locus is an assemblage of points 

 which have a common property. 



PROP. VI. Every point of the plane drawn 

 through the middle of a line perpendicular to the 

 line is equally distant from the extremities of the 

 line, and any point exterior to this plane is at 

 unequal distances from the extremities. 



Let P be a plane drawn through C, the middle 

 point of the line AB, and perpendicular to AB. 

 To prove that every point in this plane is equally 

 distant from A and B. In the plane take any point 

 whatever D. Join DA, 

 DC, DB. Then it is evi- 

 dent that the angles DC A, 

 DCB are right angles. 

 Comparing the two tri- 

 angles ACD, BCD, we 

 have two sides AC, CD 

 of the one equal to BC, 

 CD of the other, and the 

 contained angles equal, therefore AD is equal to 

 BD ; hence the point D is equally distant from 

 A and B. In the same manner it may be shewn 

 that any other point is equally distant from A 

 and B. 



Again, any point D 

 exterior to the plane is at 

 unequal distances from 

 A and B. Let the line 

 AD cut the plane in E ; 

 then AE is equal to BE. 

 Add to each ED, there- 

 fore AE + ED, or AD 

 is equal to BE, ED ; 



628 



but (Prop. VI., p. 612) BE + ED are greater than 

 BD, hence AD is greater than BD. 



Cor. Hence the plane P is the locus of ail 

 points equally distant from A and B. 



PROP. VII. If from the intersection A of the 

 perpendicular OA with the plane P a line AD be 

 drawn perpendicular to BC any line in the plane, 

 and if D be joined to any point O in OA, then 

 OD shall be perpendicular 

 to BC. In BC, take BD 

 equal to DC. Join AB, AC, 

 OB, OC. Then in the two 

 triangles ADB, ADC we 

 have two sides and the 

 contained angle in the one 

 equal to two sides and the 

 contained angle in the 

 other, therefore AB is equal to AC. Again, com- 

 paring the two triangles OAB, OAC we have 

 evidently OB equal to OC. Then in the two tri- 

 angles ODB, ODC we have the three sides of the 

 one equal to the three of the other, therefore the 

 angle ODB is equal to the angle ODC ; therefore 

 each is a right angle, and OD is perpendicular 

 to BC. 



THE LINE AND PLANE PARALLEL. 



Definition. A line is said to be parallel to a 

 plane when, being produced indefinitely, it does 

 not meet the plane. 



A plane is said to be parallel to another plane 

 when, being produced indefinitely, it never meets it 



PROP. VIII. If two straight lines are parallel, 

 every plane which is perpendicular to the one is 

 also perpendictilar to the other. 



Let AB and CD be two parallel lines, and let 

 the plane P be perpendicular to AB ; it is also 

 perpendicular to CD. Let the plane of the two 

 parallel lines AB and CD cut the plane P 

 along the straight line CB ; then as AB is per- 

 pendicular to the plane, it is 

 perpendicular to the line 

 BC, so also will CD be per- 

 pendicular to CB. Then if 

 it can be shewn that CD is 

 perpendicular to any other 

 line through C in the plane 

 P, it shall be perpendicular 

 to the plane itself. Draw in 

 the plane P, CE perpendicular to CB ; then, by 

 the last proposition, if C be joined to A, any 

 point in AB, CE is perpendicular to AC ; but it 

 was also perpendicular to CB, it is therefore per- 

 pendicular to the plane of CAB. It follows that 

 the line CD drawn through C in this plane is per- 

 pendicular to CE. But it was before shewn that 

 CD was perpendicular to CB, therefore it is per- 

 pendicular to the plane of CE and CB that is, it 

 is perpendicular to the given plane P. 



Conversely, if AB and CD be each perpendi 

 lar to the same plane P, they are parallel to each 

 other. For the line which is drawn through C 

 parallel to AB is perpendicular to the plane, and 

 must coincide with CD, as only one perpendicu' 

 to the plane can be drawn through C. 



PROP. IX. A straight line parallel to a line 

 drawn in a plane is parallel to the plane, 



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