Elements of geometry and trigonometry . 50 GEOMETRY PROPOSITION XIII. THEOREM. If the distance between the centres of two circles is equal to thesum of their radiiy the two circles will touch each other exter-nally. Let C and D be the centres at adistance from each other equal toCA+AD. The circles will evidently have thepoint A common, and they vv^ill haveno other; because, if they had twopoints common, the distance between their centres must be less than the sum of their PROPOSITION XIV. THEOREM. If the distance between the centres of two circles is equal to thedifference of their rad
Elements of geometry and trigonometry . 50 GEOMETRY PROPOSITION XIII. THEOREM. If the distance between the centres of two circles is equal to thesum of their radiiy the two circles will touch each other exter-nally. Let C and D be the centres at adistance from each other equal toCA+AD. The circles will evidently have thepoint A common, and they vv^ill haveno other; because, if they had twopoints common, the distance between their centres must be less than the sum of their PROPOSITION XIV. THEOREM. If the distance between the centres of two circles is equal to thedifference of their radii, the two circles will touch each otherinternally. Let C and D be the centres at a dis-tance from each other equal to AD—CA. It is evident, as before, that they willhave the point A common : they can haveno other; because, if they had, the greaterradius AD must be less than the sum ofthe radius AC and the distanceCD betweenthe centres (Prop. XIL) ; which is contraryto the supposition. Cor. Hence, if two circles touch each other, either exteinally or internally, their centres and the point of contact willbe in the same right line. Scholium. All circles which have their centres on the righthne AD, and which pass through the point A, are tangent toeach other. For, they have only the point A common, and itthrough the point A, AE be drawn perpendicular to AD, thestraight line AE will be a common tangent to all the circles.
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