3. To draw the radical axis of two given circles. Let A and B be the centres of the given circles. It is required to draw their radical axis. If the given circles intersect, then the st. line drawn through their points of intersection will be the radical axis. [Ex. 1, Cor. p. 397.] But if the given circles do not intersect, describe any circle so as to cut them in E, F and G, H. Join EF and HG, and produce them to meet in P. Join AB ; and from P draw PS perp. to AB. Then PS shall be the radical axis of the Os (A), (B). [The proof follows from III. 36 and Ex. 1, p. 396.] DEFINITION. If each pair of circles in a given system have the same radical axis, the circles are said to be co-axal. EXAMPLES ON THE RADICAL AXIS. 1. Shew that the radical axis of two circles bisects any one of their common tangents. 2. If tangents are drawn to two circles from any point on their radical axis ; shew that a circle described with this point as centre and any one of the tangents as radius, cuts both the given circles ortho. gonally. 3. O is the radical centre of three circles, and from O a tangent OT is drawn to any one of them : shew that a circle whose centre is O anıl radius OT cuts all the given circles orthogonally. 4. If three circles touch one another, taken two and two, shew that their common tangents at the points of contact are concurrent. 5. If circles are described on the three sides of a triangle as diameter, their radical centre is the orthocentre of the triangle. 6. All circles which pass through a fixed point and cut a given circle orthogonally, pass through a second fixed point. 7. Find the locus of the centres of all circles which pass through a given point and cut a given circle orthogonally. 8. Describe a circle to pass through two given points and cut a given circle orthogonally. 9. Find the locus of the centres of all circles which cut two given circles orthogonally. 10. Describe a circle to pass through a given point and cut two given circles orthogonally. 11. The difference of the squares on the tangents drawn from any point to two circles is equal to twice the rectangle contained by the straight line joining their centres and the perpendicular from the given point on their radical axis. 12. In a system of co-axal circles which do not intersect, any point is taken on the radical axis ; shew that a circle described from this point as centre, with radius equal to the tangent drawn from it to any one of the circles, will meet the line of centres in two fixed points. [These fixed points are called the Limiting Points of the system.] 13. In a system of co-axal circles the two limiting points and the points in which any one circle of the system cuts the line of centres form a harmonic range. 14. In a system of co-axal circles a ing point has the same polar with regard to all the circles of the system. 15. If two circles are orthogonal any diameter of one is cut harmonically by the other. In the two following theorems we are to suppose that the segments of straight lines are expressed numerically in terms of some common unit; and the ratio of one such segment to another will be denoted by the fraction of which the first is the numerator and the second the denominator. 2c H.S.E. DEFINITION. A straight line drawn to cut a given system of lines is called a transversal. a 1. If three concurrent straight lines are drawn from the angular points of a triangle to meet the opposite sides, then the product of three alternate segments taken in order is equal to the product of the other three segments. F B D CB C D Let AD, BE, CF be drawn from the vertices of the ' ABC to intersect at O, and cut the opposite sides at D, E, F. Then shall BD.CE. AF =DC. EA. FB. Now the As AOB, AOC have a common base AO; and it may be shewn that BD: DC=the alt. of A AOB : the alt. of AOC; BD A AOB СЕ ДВОС similarly, EA A BOA : AF ДСОА and FB ДСОВ' Multiplying these ratios, we have BD CE AF DC EA FB or, BD. CE . AF =DC. EA.FB. Q.E.D. A AOC; = =l; . Note. The converse of this theorem, which may be proved indirectly, is very important: it may be enunciated thus : If three straight lines drawn from the vertices of a triangle cut the opposite sides so that the product of three alternate segments taken in order is equal to the product of the other three, then the three straight lines are concurrent. That is, if BD. CE . AF =DC. EA. FB, then AD, BE, CF are concurrent. 2. If a transversal is drawn to cut the sides, or the sides produced, of a triangle, the product of three alternate segments taken in order is equal to the product of the other three segments. Let ABC be a triangle, and let a transversal meet the sides BC, CA, AB, or these sides produced, at D, E, F. Then shall BD. CE. AF =DC. EA.FB. BD HA FB AF: and from the similar As DCE, HAE, CE EA DC HA BD CE EA .., by multiplication, FB'DC-AF that is, BD. CE. AF =1, DC. EA. FB or, BD.CE. AF =DC. EA. FB. Q.E.D. : NOTE. In this theorem the transversal must either meet two sides and the third side produced, as in Fig. 1; or all three sides produced, as in Fig. 2. The converse of this theorem may be proved indirectly : If three points are taken in two sides of a triangle and the third side produced, or in all three sides produced, so that the product of three alternate segments taken in order is equal to the product of the other three segments, the three points are collinear. DEFINITIONS. 1. If two triangles are such that three straight lines joining corresponding vertices are concurrent, they are said to be co-polar. 2. If two triangles are such that the points of intersection of corresponding sides are collinear, they are said to be co-axial. The propositions given on pages 111-114 relating to the con. currence of straight lines in a triangle, may be proved by the method of transversals, and in addition to these the following important theorems may be established. THEOREMS TO BE PROVED BY TRANSVERSALS. 1. The straight lines which join the vertices of a triangle to the points of contact of the inscribed circle (or any of the three escribed circles) are concurrent. 2. The middle points of the diagonals of a complete quadrilateral are collinear. [See Def. 4, p. 387.] 3. Co-polar triangles are also co-axial ; and conversely co-axial triangles are also co-polar. 4. The six centres of similitude of three circles lie three by three on four straight lines. |