meet at I, and perpendiculars ID1, I11, IF1 be drawn to the sides BC, and AC, AB produced, it may be proved that ID1, IE1, IF1 are all equal, and .. that I is the centre of a circle touching BC, and AC, AB produced. Hence also, I, the point of intersection of the bisectors of the exterior angles at C and A, will be the centre of a circle touching CA, and BA, BC produced; I, the point of intersection of the bisectors of the exterior angles at A and B, will be the centre of a circle touching AB, and CB, CA produced. COR.-The following sets of points are collinear: A, I, I1; B, I, I1⁄2 ; C, I, I3; 12, A, I3; I3, B, I1; I1, C, I1⁄2. In other words, the six bisectors of the interior and exterior angles at A, B, C meet three and three in four points, I, I1, I„, I ̧, which are the centres of the four circles touching the three given straight lines. Or, the six straight lines joining two and two the centres of the four circles which touch AB, BC, CA, pass each through a vertex of the A ABC. The circles whose centres are I, I, Is are called escribed or exscribed circles of the ▲ ABC, an expression which, in its French form (ex-inscrit), is said to be due to Simon Lhuilier. See his Élémens d' Analyse Géométrique et d'Analyse Algébrique (1809), p. 198. It is usual to denote the radius of the circle inscribed in a triangle by r, the radii of the three escribed circles by 71, 72, 73, and the radius of the circumscribed circle by R. THE MEDIOSCRIBED CIRCLE. PROPOSITION 2. The circle which passes through the middle points of the sides of a triangle passes also through the feet of the perpendiculars from the vertices to the opposite sides, and bisects the segments of the perpendiculars between the orthocentre and the vertices. Let ABC be a triangle; H, K, L the middle points of its sides ; X, Y, Z the feet of its perpendiculars; U, V, W the middle points of AO, BO, CO: it is required to prove that one circle will pass through these nine points. Join HK, HL, HU, HV, HW, KU, KV, UV, UW, WL, UL. In ▲ ABO, LU is || BO, and in ▲ CBO, HW is || BO; LU is || HW. Similarly, in As ABC, AOC, LH and UW are || AC; App. I. 1 .. LHWU is a m But since BO is 1 AC, .. LU, HW are 1 LH, UW; .. LHWU is a rectangle. App. I. 1 the four points H, W, U, L lie on the circle described with HU or LW as diameter. Similarly, HK, UV are || AB, and HV, KU || CO; and since AB is 1 CO, .. HKUV is a rectangle. III. 31 App. I. 1 .. the four points H, K, U, V lie on the circle described with HU or KV as diameter. III. 31 Hence the six points H, K, L, U, V, W lie on the same circle, and HU, KV, LW are diameters of it. But since the angles at X, Y, Z are right; .. X lies on the circle whose diameter is HU, COR. 1.-Since HU, KV, LW are diameters of the same circle, their common point of intersection M is the centre. COR. 2.-M is midway between the orthocentre and the circumscribed centre. Let S be the circumscribed centre, and SH be joined. Then SH is BC (III. 3); and .'. || OU. But SH OU (App. I. 5, Cor.); .. SHOU is a m; .. the diagonal SO bisects HU, that is, passes through M, and is itself bisected at M. COR. 3.-The medioscribed diameter = For SHUA is a m; and .. HU = SA. circle. the circumscribed radius. ▲ 8 ABC, AOB, BOC, COA have the same medioscribed Since the medioscribed circle of ▲ ABC passes through U, L, V, the middle points of the sides of ▲ AOB, and since a circle is determined by three points; .. the medioscribed circle of ▲ ABC must also be the medioscribed circle of ▲ AOB. Similarly for the other triangles. COR. 5.-By reference to Cor. 3, it will be seen that the circles circumscribed about As ABC, AOB, BOC, COA must be equal. (Carnot, Géométrie de Position, 1803, § 130.) COR. 6.-The medioscribed circle of A ABC is also the medioscribed circle of an infinite series of triangles. For H, K, L, the middle points of the sides of ▲ ABC, may be taken as the feet of the perpendiculars of another ▲ A'B'C'; the middle points of the sides of ▲ A'B'C' may be taken as the feet of the perpendiculars of a ▲ A"B"C"; and so on. Or, instead of the median ▲ HKL, ▲s XKL, YLH, ZHK may be taken as median triangles, and the triangles formed of which they are the median triangles; and so on. [The circle HKL is generally called the nine-point circle of ▲ ABC, a name given by Terquem, 'le cercle des neuf points.' Following, however, the suggestion of an Italian geometer, Marsano, who calls it 'il circolo medioscritto,' I have adopted the name medioscribed. The property that one circle does pass through these nine points was first published in Gergonne's Annales de Mathématiques, vol. xi. p. 215 (1821), in an article by Brianchon and Poncelet. See this reference, or Poncelet's Applications d'Analyse et de Géométrie, vol. ii. p. 512. It is probable that K. W. Feuerbach of Erlangen, and T. S. Davies of Woolwich, also discovered the property independently, though they were later in publication. See Feuerbach's Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks (1822), and a paper by Davies on 'Symmetrical Properties of Plane Triangles,' in the Philosophical Magazine for July 1827. For other proofs, see Rev. Joseph Wolstenholme in Quarterly Journal of Pure and Applied Mathematics, vol. ii. pp. 138, 139 (1858); W. H. Besant in the Messenger of Mathematics (old series), vol. iii. pp. 222, 223 (1866); William Godward in the Mathematical Reprint from the Educational Times, vol. vii. p. 86 (1867); Desboves' Questions de Géométrie Élémentaire, 2ème éd. p. 146 (1875); and Casey's Elements of Euclid, p. 153 (1882). The proof in the text was given by T. T. Wilkinson of Burnley in the Lady's and Gentleman's Diary for 1855, p. 67. It may be mentioned that it was discovered by Feuerbach (see his Eigenschaften, &c. § 57) that the medioscribed circle touches the inscribed and escribed circles of ▲ ABC. The proofs that have been given of this theorem by elementary geometry are rather complicated: see Lady's and Gentleman's Diary for 1854, p. 56; Quarterly Journal of Pure and Applied Mathematics, vol. iv. (1861), p. 245, and vol. v. (1862), p. 270; Baltzer, Die Elemente der Mathematik, vol. ii. pp. 92, 93. It is also proved by J. J. Robinson in the Lady's and Gentleman's Diary for 1857 (and it seems to have been first noted by T. T. Wilkinson), that the medioscribed circle touches an infinite series of circles.] DEDUCTIONS. 1. Every equilateral figure inscribed in a circle is equiangular. 2. In a given circle inscribe (a) three, (b) four, (c) five, (d) six equal eircles touching each other and the given circle. 3. The perpendicular from the vertex to the base of an equilateral the side of an equilateral triangle inscribed in a = triangle 4. The area of an inscribed regular hexagon = three-fourths of the area of the regular hexagon circumscribed about the same circle. 5. Inscribe a regular hexagon in a given equilateral triangle, and compare its area with that of the triangle. 6. Inscribe a regular dodecagon in a given circle, and prove that its area= that of a square described on the side of an equilateral triangle inscribed in the same circle. 7. Construct a regular octagon on a given straight line. 8. A regular octagon inscribed in a circle = the rectangle contained by the sides of the inscribed and circumscribed squares. 9. The following construction is given by Ptolemy (about 130 a.d.) in the first book of his Almagest, for inscribing a regular pentagon and decagon in a circle: Draw any diameter AB, and from the centre draw CD 1 AB, meeting the Oce at D; bisect AC at E, and join ED. From EB cut off EF = ED, and join DF. CF will be a side of the inscribed regular decagon, and DF a side of the inscribed regular pentagon. Prove this. 10. A ribbon or strip of paper whose edges are parallel, is folded up into a flat knot of five edges. Prove that the sides of the knot form a regular pentagon. 11. Construct a regular decagon on a given straight line. 12. In a given square inscribe an equilateral triangle one of whose vertices may be (a) on the middle of a side, (b) on one of the angular points, of the square. Construct a triangle having given 13. The inscribed circle, and an escribed circle. 14. Two escribed circles. 15. Any three of the centres of the four contact circles. |