A first course in projective geometry . Fig. 19. AX PAThen — = —, and the angle AXP = angle BXQ; .. theXB QB triangles, being such that the angles at A and B are both obtuse—as in the figure—or both acute, are similar; . PX PA , ^ • • ^:77^ = T^Tr^ = QB SIMILITUDE 47 Similarly we could prove the property for any secantthrough Y Def. The points X, Y are called the centres of similitudeof the two circles. They are clearly the points where the direct and inversecommon tangents cut the line of centres. For let the direct common tangent PQ cut the line ofcentres at Y. Since AP is parall
A first course in projective geometry . Fig. 19. AX PAThen — = —, and the angle AXP = angle BXQ; .. theXB QB triangles, being such that the angles at A and B are both obtuse—as in the figure—or both acute, are similar; . PX PA , ^ • • ^:77^ = T^Tr^ = QB SIMILITUDE 47 Similarly we could prove the property for any secantthrough Y Def. The points X, Y are called the centres of similitudeof the two circles. They are clearly the points where the direct and inversecommon tangents cut the line of centres. For let the direct common tangent PQ cut the line ofcentres at Y. Since AP is parallel to BQ, the triangles APY,BQY are similar. Hence AY/BY = AP/BQ = AY/BY. .*. Y and Y must coincide. Similarly for the inverse common tangents. § 9. Prop. If a variable circle touch two fixed circles,the line joining the points of contact passes through one orother of two fixed points (the centres of similitude of thefixed circles). Let A, B (Fig. 20) be the centres of the fixed Pio. 20. Let PQ be the chord of contact of the variable circle whosecentre is C. Then CQ and CP pass through A and B respec-tively. Let PQ cut the circle, whose centre is A, again in P. Then angle APQ = angle AQP = angle CQP = angle CPQ. 48 PROJECTIVE GEOMETRY .. AP is parallel to CPB, and the triangles APY, BPY are similar, so that — = —, and Y is the external centre ofsimilitude. The student should draw figures for the other cases of thisproposition in which the variable circle is touched internallyby either one or both of the fixed circles. The number ofsuch is increased by including the cases in which one or otherof the fixed circles reduces to a point ( has zero radius), orhas infinite radius { becomes a straight line). The determination of the centres of similitude in thesespecial cases is a useful exercise. One such will be considered here. The centres of similitude of a circle and a straight line arethe ends of the diameter of the circle perpendicular to thestr
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