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Let AO, AO' be two antiparallels with respect to A, so that BAO = 4CAO'. On AB drop the perpendiculars Oz, O'Z'; and on AC, the perpendiculars OY, O'Y'. Then, since a circle goes round both AZOY and AZ'O'Y',
:. <ZY0= L ZAO = 2 Y'AO' = _ Y'Z'O'. And the angles at 0, 0' are both equal to the supplement of A. .. the A's OZY, O'Y'Z' are similar.
.:. OZ.O'Z' = OY.O'Y'. Similarly if BO, BOʻ are antiparallels with respect to B, and 0X, O'X' perpendiculars on BC, OX.O'X'=OZ.O'Z',
.. OX.O'X' = OY.O'Y'. .. the a's OXY, O'Y'X' are similar.
.. . XCO = XYO =_Y'X'O' = _ Y CO'. .. OC, O'C are antiparallels with respect to C.
Also, if T is the middle point of 00', because OX, O'X' are perpendiculars on BC,
.. TX = 0T2+ OX. O'X'. .. TX=TX' = TY=TY' = TZ=TZ'. .. the points XX'YY' ZZ' lie on a circle whose centre is T.
In particular taking 0 and O' to be the orthocentre and circumcentre, we have the nine-points circle. Similar propositions hold for the gravity centre and cosine centre, and for the two Brocard Points.
[The following are easy examples, which may be regarded as of the nature of book-work.]
1. If BC, B'C' be antiparallel with respect to the angle at A, ABC and AB'C" are similar triangles.
2. If A, B, C, D be four points in any order on the circumference of a circle, AB, CD shall be antiparallel with respect to the angle between AC and BD.
3. If the bisector of A cuts BC in H, the antiparallel of BC (with respect to A) through H will cut off from AB and AC parts equal to AC and A B respectively.
4. The tangent to the inscribed circle from either vertex of the triangle is equal to half the excess of the sum of the sides through the vertex over the base.
5. The tangent to the escribed circle from the opposite vertex is equal to half the sum of the sides of the triangle.
6. The external common tangent of the inscribed and escribed circles is equal to the side of the triangle, to which the escribed circle is exterior.
7. The internal common tangent of the inscribed and escribed circles is equal to the difference between the two sides to which the escribed circle is interior.
8. The fourth common tangent to the inscribed and escribed circles forms with the two sides a triangle equal in all respects to the primitive triangle.
9. The distance between the middle point of a side and its point of contact with the inscribed circle is equal to half the difference between the two other sides.
10. If the sum of two opposite sides of a quadrilateral is equal to the sum of the two other opposite sides, a circle may
be inscribed in the quadrilateral.
The angle which each side subtends at the incentre exceeds half the opposite vertical angle by a right-angle; and that which each side subtends at an ecentre is the complement of half the opposite angle.
12. The incentre is the orthocentre of the triangle formed by the ecentres.
13. The lines joining the angle of a triangle to the extremities of that diameter of the circumcircle which bisects the base are the internal and external bisectors of the angle.
14. The distance between the centres of the inscribed and either escribed circle is bisected by the circumcircle.
15. The rectangle of the segments of any chord of the circumcircle drawn through the centre of an escribed circle is equal to twice the rectangle of their radii.
16. The feet of the perpendiculars on the sides from any point on the circumcircle lie on a straight line.
17. The lines joining the vertex to the circumcentre and orthocentre are inclined at an angle equal to the difference between the base angles.
18. If OL, the perpendicular from the orthocentre on the side BC, is produced to meet the circumcircle in O', OL=O'L.
= . 19. The angles of the Pedal triangle (i.e. the triangle formed by joining the feet of the perpendiculars from the opposite angles) are supplementary respectively to the doubles of those of the primitive triangle.
20. If a triangle is formed whose sides are, with respect to the angles of ABC, antiparallels to the sides; and another formed from this in the same way, and so on, the angles of the nth triangle so formed will be 60° + (-2)" (A – 60°), 60° +(-2)" (B - 60°), 60° + (-2)» (C – 60°).
21. The angle which each side subtends at the orthocentre is the supplement of the opposite vertical angle.
22. If, in a given triangle, another triangle has to be inscribed whose sides shall be parallel respectively to those of the
given triangle, then the vertices of the inscribed triangle must be the middle points of the sides of the given triangle.
23. If, in a given triangle, another triangle has to be inscribed whose sides shall be antiparallel respectively to those of the given triangle, then the vertices of the inscribed triangle must be the feet of the perpendiculars on the sides of the given triangle.
24. The centre of gravity of the triangle DEF is the same as that of ABC, and its orthocentre is the circumcentre of ABC.
25. If the middle points of the sides of a triangle be joined, and the middle points of the sides of the triangle so formed be joined, and so on; then all the triangles so formed will have the same centre of gravity, and the circumcentre of each will be the orthocentre of the next.
26. The three triangles into which the centre of gravity divides a triangle are each one-third of the triangle.
27. The triangles cut off by joining the middle points of the sides are each one quarter of the primitive triangle.
28. The lines joining the middle points of the sides with the middle points of the perpendiculars from the opposite angles intersect at the cosine-centre.
29. The lines joining the vertices of a triangle to its points of contact with the inscribed circle intersect at the cosine-centre of the triangle formed by those points of contact.
30. The triangles into which the cosine-centre divides a triangle are to one another as the squares on the sides.
The line joining the vertex of the triangle to the cosinecentre divides the base into segments which are to one another as the squares on the other sides.
32. The radius of the cosine-circle is to that of the circumcircle as the perpendicular from the cosine-centre on either side is to half that side.
33. The distance of a vertex from the cosine-centre is to its distance from the middle point of the opposite side as the cosinediameter is to the opposite side.
34. The cosine-centre is the centre of gravity of the triangle formed by the feet of the perpendiculars from it upon the sides.
35. The line joining the circumcentre of the three ecentres to the incentre of a triangle is bisected by the circumcentre.
36. If C be taken on AB so that AC" AC, and if the bisector of A cut CC' in D' and BC in H, then DD' is the mean proportional between DH and DL.
Show how, from the last example, it follows that the ninepoints circle and inscribed circle touch.
37. If BC2, C,A,, A,B, be any three diameters of a circle and a triangle be formed by joining the extremities A, A2, B,B, C,C2, then the circle is the cosine-circle of the triangle so formed.
38. Show that, by the method of the last example, eight different triangles may be described with the same three diameters: and that, taking any two of these triangles, either all the sides of the one are parallel or one is parallel and the other two perpendicular to the sides of the other.
39. In Lemoine's circle the three chords of intersection opposite to the three parallels to the sides are antiparallels to the sides and each is equal to the cosine-radius.
40. Each of the six triangles into which the Lemoine Hexagon is divided by joining its vertices to the cosine-centre is similar to the primitive triangle.
41. The three segments into which the Lemoine circle cuts each side are to one another as the squares on the three sides.
42. The circles, whose intersection gives one of the Brocard Points, may be defined as passing through one vertex and touching the opposite side at one of its extremities. 43. If any points a, B, y be taken on the sides BC, CA, AB
γ of a triangle, the circumcircles of Aßy, Bya, Caß will cut in a single point.