GEOMETRY. 



(2.) If the line OC be drawn 

 from the centre to C, the 

 middle point of AB, OC shall 

 be perpendicular to AB. Join 

 OA, OB. In the two triangles 

 OCA, OCB we have the three 

 sides of the one equal to the 

 three sides of the other, there- 

 fore (Prop. VII. 3) their angles 

 are equal namely, the angle ACO to the angle 

 BCO ; but these are adjacent angles, therefore 

 OC is perpendicular to AB. 



Definition. A tangent to the circle is a chord 

 which cuts it in two points infinitely near each 

 other, and either of these points is called the 

 point of contact. 



Hence it follows from the above : 



(i.) That the line which is drawn from the 

 point of contact perpendicular to the tangent 

 passes through the centre. 



(2.) A line drawn from the centre perpendicular 

 to the tangent passes through the point of contact. 



(3.) A line drawn from the centre to the point 

 of contact is perpendicular to the tangent. 



CONSTRUCTIONS (ill.). 



(i.) Through a given point to draw a tangent 

 to the circle. 



(a.) If the given point be 

 on the circumference. Let 

 P be the given point 

 through which it is re- 

 quired to draw a tangent 

 to the circle. Join P with 

 O the centre, and through 

 P draw SPT perpendicular 

 to OP. ST is evidently the s 

 tangent required. 



There can only be one tangent drawn through 

 the point P ; for if there could be another, let it 

 be MN. Then since MN 

 is a tangent, the angle 

 OPN is a right angle ; but 

 the angle OPT is also a 

 right angle, therefore the 

 angle OPN is equal to the 

 angle OPT, which is ab- 

 surd. Hence, when the 

 point is on the circumfer- 

 ence of the circle, there 

 can only be one tangent drawn through it to the 

 circle. 



(.) If the given point be without the circum- 

 ference. Let P, as before, be the given point. 

 From P as centre, with 

 radius equal to PO, 

 describe a circle ; and 

 from O as centre, with 

 radius equal to the 

 diameter of the given 

 circle, describe another 

 circle cutting the circle 

 whose centre is P in M 

 and N. Join OM and 

 ON, and let these lines 

 cut the given circle in 



S and V. Join PS and '*" 



PV. The lines PS and 



PV are lines drawn from P touching the given 



circle. This is evident, for the lines OM and ON 



are bisected in S and V, and these points are 

 joined with P, the -centre of the circle ; therefore 

 each of the angles PSO, PVO is a right angle, 

 therefore PS and PV are tangents. 



PROP. XIII. (i.) The diameter is the greatest 

 chord in a circle. 



(2.) Of others, those are equal which are equally 

 distant from the centre, and 

 conversely. 



(i.) Let AB be the dia- 

 meter of a circle whose 

 centre is O ; it is required 

 to prove that it is greater 

 than a chord CD, which is 

 not a diameter. Join CO, 

 DO. Then the two sides 

 CO and DO are together greater than CD (Prop. 

 VI.) ; but CO and DO are equal to the diameter 

 AB, therefore AB is greater than CD. 



(2.) Let AB and CD be two chords which are 

 equally distant from the centre that is, such that 

 the perpendiculars OQ and 

 OP upon them from the 

 centre are equal ; it is easily 

 proved (Prop. VII. 5) that 

 AQ is equal to PD, and 

 therefore the doubles of these, 

 or AB and CD, are also 

 equal. 



The converse proposition 

 is evident. 



Intersection and Contact of two Circles. 



PROP. XIV. If two circles intersect each other, 

 the line which joins their centres bisects perpen- 

 dicularly their common chord. 



Let the two circles whose centres are O and Q 

 intersect in A and B ; it is required to prove that 

 the line which joins O and Q bisects the line AB 

 at right angles. Join O and Q with A and B. 

 Comparing the two 

 triangles AOQ,BOQ 

 we have the three 

 sides in the one tri- 

 angle equal to the 

 three sides in the 

 other, therefore 

 (Prop. VII. 3) the 

 angle AOQ is equal 

 to the angle BOQ. Again, comparing the two 

 triangles AOC, BOC we have the two sides AO, 

 OC in the one equal to the two sides BO, OC 

 in the other, and the contained angle AOC equal 

 to the contained angle BOC, therefore (Prop. VII. 

 i) the side AC is equal to the side BC, and the 

 angle ACO is equal to the angle BCO ; but these 

 are adjacent angles, therefore each is a right 

 angle, and OQ is perpendicular to AB. 



Cor. If the circle O remain fixed, and the 

 circle Q be sup- 

 posed drawn out- 

 wards, so that the 

 points of intersec- 

 tion A and B grad- 

 ually approach 

 each other and ul- 

 timately coincide, 

 the common chord 

 becomes the com- 

 mon tangent, and the middle point C of this 



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