2o. The segments AA', BB', CC' have a common segment of harmonic section. [To prove 1°. By hyp. since [CABA']=[C'A'B'A]; on rearranging, by Ex. 1, we get [AA'BC]=[A'AB'C']=[AA'C'B']. Therefore by alternation (Ex. 5) [AA'BC']=[AA'CB']=[A'AC'B]; similarly for all other combinations. To prove 2°. Let MN divide the segments AA' and BB' harmonically, it divides CC' also harmonically. For [MABA']=[MA'B'A] (by Ex. 7) and [NABA']=[NA'B'A]; also by 1° [CABA']=[C'A'B'A] and [C'ABA']=[C'A'B'A], hence (Ex. 6) [MNCC']=[MNC'C]; therefore, etc. (Ex. 2)]. 10. Show generally for two equianharmonic systems if any two conjugates A and A' are interchangeable, e.g., if [ABCD]=[A'B'C'D] and [A'BCD]=[AB'C'D'] that 1o. Every four are equianharmonic with their four opposites; 2o. The segments AA', BB', CC', DD' have a common segment of harmonic section. [By the method of Ex. 9.] SECTION II. ANHARMONIC SECTION OF AN ANGLE. 134. It has been explained in Art. 3 that the anharmonic ratio of four points A, B, C, D is equal to that of the pencil O. ABCD formed by joining them to any point O. It follows then that all the properties of four collinear points stated in the previous section involve correlative properties of a pencil of rays, and that the latter are immediately derived from the former by aid of the equation BC. AD: CA.BD:AB.CD = sin BC. sin AD: sin CA. sin BD: sin AB. sin CD. Also by describing a circle through the vertex O of the pencil O. ABCD, and denoting by A, B, C, D the points where it meets the legs of the pencil again; since the sines of the angles at O are in the ratios of the chords opposite to them we may further obtain from the anharmonic properties of collinear points corresponding relations amongst points which lie on a circle. 135. The following properties will appear evident :1o. All transversals to a pencil of rays are cut equianharmonically. 2o. A transversal to a pencil drawn parallel to one of its rays D is divided by the remaining three in the simple ratio AC/BC; which is the anharmonic ratio of the pencil. 3o. In 2°, if the pencil is harmonic, any transversal A'B'C' parallel to D is such that A'B'B'C'. 4°. For any two equianharmonic rows of points A, B, C, D, ... and A', B, C, D', ..., if the lines AA', BB', and CC are concurrent at 0; DD' and all other lines joining corresponding points of the given systems pass through O. [This important property is the converse of 1° and follows easily by an indirect proof.] 136. Theorem.-If two lines be divided equianharmonically such that a pair of corresponding points coincide at their intersection [OABC...]=[OA'B'C'...] the systems are in perspective; and reciprocally if two equianharmonic pencils are such that a pair of corresponding rays coincide on the lines joining their vertices they are in perspective. Let AA' and BB' meet in P. let PC cut the other axis in C". Join PC, and if possible [OABC]=[OA′BC], since the rows are in perspective. But [OABC]=[OA′B′C] (hyp.); therefore [OA'B'C']=[OA'B'C'], i.e. C and C" coincide. Reciprocally for any two pencils P. ABC, ... and P'. A'B'C', ... if the rays A, A' and B, B' intersect respec tively in X and Y, it follows that C and C meet on the line XY. Otherwise thus:-The rows [XYZW] and [XYZ' W] are equianharmonic; therefore Z and Z' coincide. COR. 1. If two pencils are equianharmonic, any two rows passing through the intersection of a pair of corresponding rays are in perspective. COR. 2. Through a given point P a line may be drawn across a triangle ABC, cutting its sides in the points Q, R, S, such that [PQRS]=a given anharmonic ratio. [For the pencil (A. PQRS) formed with the row at any vertex A of the triangle is given, and since three of its rays are given the fourth is known.] Def. Lines divided equianharmonically are also said to be divided Homographically. The term homographic is applied in general to the equianharmonic division of figures of the same kind, e.g. lines, circles, etc., etc. EXAMPLES. 1. Every tangent to a circle is cut harmonically by the sides of the escribed square. [In the limiting position when the variable tangent coincides with a side of the square the row of points determined on it are harmonic; therefore, etc., Art. 81, Ex. 3.] 2. To express the anharmonic ratios in which a variable tangent is divided by four fixed tangents, in terms of the chords of contact of the tangents. [Let P, Q, R, S denote the points of contact of the sides of the escribed quadrilateral, which meet the variable tangent at 0 in A, B, C, D; O' the centre of the circle. Then ABCD=0'. ABCD =0. PQRS, since O'A, OP; O'B, OQ... are four pairs of perpendicular lines; therefore the required expressions are QR. PS: RP. QS: PQ. RS.] 3. For any quadrilateral escribed to a circle at the points P, Q, R, S, each pair of diagonals and a corresponding pair of opposite connectors of the inscribed quadrilateral PQRS are concurrent. (See Art. 67, Cor. 8.) [To prove that the sets of lines QR, PS, YY', ZZ' RP, QS, ZZ', XX' PQ, RS, XX', YY' are each concurrent. Consider each of the four tangents at the points P, Q, R, S a transversal to the quadrilateral XX'YY'ZZ'. Since consecutive tangents meet on the circle, the tangents at P and Q are cut in the same order at the points P, Z, Y, X' and Z, Q, X, Y'; therefore [PZYX']=[ZQXY']=[QZY'X]. Hence PQ, YY', XX' are concurrent. Similarly RS, YY' and XX' are concurrent; therefore, etc.] NOTE.-As the above properties are more generally true for the Conic, we consider an interesting case which arises in the parabola when the fourth tangent is at infinity (Art. 81). Let tangents AC and BC be drawn to a parabola at the points A and B, and a third tangent XY meeting BC and CA in X and Y respectively. Then the equianharmonic relations easily reduce to BX/CX=CY/AY; or a variable tangent divides two fixed tangents in the same ratio. It also subtends a constant angle at the focus. Therefore the foci of the three parabolas described to touch each pair of sides (b, c, etc.) of a triangle ABC at the extremities of the third side (BC) are the vertices of Brocard's second triangle. 4. If a circle touch four others the anharmonic ratios of the points of contact are equal to 23. 14:31.24 : 12.34. 5. The anharmonic ratios of the points of contact of the ninepoints-circle with the in- and three ex-circles of the triangle ABC are a2-b2 b2- c2 c2- a2 a2 - c29 b2- a2, c2 — b2 6. If the anharmonic ratios of four points A, B, C, D on a circle (or conic) be denoted by λ, μ, v, etc., to prove that the anharmonic ratios of the pencil P. ABCD are λ2, μ2, v2, etc., where P is the pole of the line AB. [Let PC, PD meet the conic again in C', D', and AB in E, G ; then CD', DC', and AB are concurrent at F; and since C. ABCD D. ABCD, [ABCD]=[ABEF]=[ABFG]=λ (say); therefore = |