EXERCISES ON CO-AXAL CIRCLES. 1. The circle of similitude of two circles is co-axal with them. 2. The three circles of similitude of three circles, taken two and two, are co-axal. 3. Given two intersecting circles A, B; show that there is another circle C, co-axal with them, such that, if tangents are drawn to the three from any point, then sq. on tang. to A+ sq. on tang. to B = 2 sq. on tang. to C. 4. A variable circle cuts two fixed circles orthogonally; find the Locus of its centre. 5. If a variable circle touches two fixed circles, its radius bears a constant ratio to the distance of its centre from their radical axis. 6. A variable circle goes through a fixed point A, and cuts a fixed circle orthogonally in P, Q: show that the rectangle under AP, AQ varies as the chord PQ. NOTE-Use Theorem (2), Cor. 2, and vi. Addenda (8). 7. Given a fixed circle, centre C, and a fixed line AB; if a system of circles have AB as their central axis, and cut given circle orthogonally, they are co-axal, their radical axis being the perpendicular from C on AB. 8. In a co-axal system, of the limiting point species, if through a limiting point L, a line is drawn to cut a circle of the system in X, Y; and if XM, YN are perpendiculars on the radical axis, then XM. YN LA2, where A is the intersection of the radical and central axes. = 9. Perpendiculars drawn through the mid points of the sides of a triangle to the bisectors of its angles, are the radical axes of the in-circle and the several ex-circles. 10. X, Y, P, Q, R are any five circles; if ABC is the triangle formed by the radical axes of X, P, of X, Q, and of X, R; and aßy is the triangle formed by the radical axes of Y, P, of Y, Q, and of Y, R; then ABC, aßy are in perspective. 11. If AOD, BOE, COF are the altitudes of a triangle ABC, and G its centroid; then the circum-circles of ADG, BEG, CFG, have a second point in common; namely, the intersection of OG with the radical axis of the N. P. circle and the circum-circle of ABC. 12. The sixteen circles of contact, of the four triangles formed by four intersecting lines, have their centres, in fours, on four co-axal circles. SECTION v-"THE TANGENCIES." Given in position any three of the following nine-three points, three lines, three circles of fixed magnitude-it is required to describe a circle which shall pass through such points as are given, and touch such lines and circles as are given. These Problems are known as The Tangencies. There are ten cases. Case 1. Given three points-Euc. iv. 5. Case 2. Given three lines-Euc. iv. 4. Case 3. Given two points and a line. Р Let A, B be given pts. and XY given line. Join AB, and let it meet XY in C. In XY take CP a mean propl. to CA, CB; so that CP2 = CA. CB. Then the thro. A, B, P touches XY at P. There are two positions of P, and .. two solutions. Case 4. Given two lines and a point. Let OX, OY be the given lines; A the given pt. Draw ON the bisector of XOY; and let B be the image of A with respect to ON. By Case 3, draw a O thro. A, B to touch OX in P: then this will clearly touch OY in Q, the image of P with respect to ON. There are two positions of P, and .. two solutions. A a Case 5. Given two points and a circle. Let A, B be the given pts. Draw any thro. A, B to cut given O in C, D. Let AB, CD meet in O; and draw OP to touch given O in P. Then, since OP2 .. O thro. A, B, P touches OP in P; and also touches given O in P. There are two positions of P, and .. two solutions. Case 6. Given two circles and a point. K B Let A be given pt.; and B, C centres of given Os. Join SA; and in it take SX the fourth propl. to SA, SK, SL, so that Thro. A and X describe (Case 3) O to touch O B in P; let O be its centre. Join SP, cutting O B in D again, and meeting O C. in Q. Then SA. SX = SK.SL = SP. SQ, . S is cent. of simil. .. Q is on thro. A, X, P. Also BD is || to CQ, by a property of centre of similitude. Since there are two positions of P, there are two solutions corresponding to S. Also taking the internal centre of similitude there will be two more solutions. i. e. there are four solutions in all. Case 7. Given a point, a line, and a circle. B N Let A be given pt.; C centre of given ; and BD given line. Draw CN to BD at N, and let it meet in F. Produce NC to meet again in E. Join EA; and in it take EX the fourth propl. to EA, EN, EF; so that Thro. A and X describe (Case 3) a O to touch BD in Q. Then since FPE and FNQ are each right, .. quad. FNQP is cyclic. Two Os can be described thro. A and X, touching BD, which give two solutions; and similarly two more solutions can be got by joining FA, instead of EA. Case 8. Given two lines, and a circle. B M Let AB, DE be given lines; C centre of given O. On sides of AB, DE, remote from C, draws to them, at distance from them which rad. of C. = Describe (Case 4) a O thro. C to touch the | to AB in M, and the || to DE in N. Let O be its centre; and let OM cut AB in Q, and ON cut DE in R. Join CO cutting given O in P; and with centre O and radius OP describe a O. Also Q and R are pts. on OO. touches AB, DE at these pts., . As there are right. The two solutions of Case 4 give two solutions: two more will be got by drawing |s to AB, ED on same side as C. |