B F R Р А E H ::: LOXB= 7, the on OB as diameter passes through X. (107°, Cor. 1) Similarly the O on OC as diameter passes through X. Therefore the Os on OB and OC intersect in X; and in like manner it is seen that the Os on OC and OA intersect in Y, and those on OA and OB intersect in Z, the foot of the I from 0 to AB. Ex. 6. The feet of the medians and the feet of the altitudes in any triangle are six concyclic points, and the circle bisects that part of each altitude lying between the orthocentre and the vertex. D, E, F are the feet of the medians, i.e., the middle points of the sides of the AABC. Let the circle through D, E, F cut the sides in G, H, K. Now FD is || to AC and ED is || to AB, (84", Cor. 2) .. LFDE=LFAE. But LFDE=LFHE, (106°, Cor. 1) :: AAFH is isosceles, and AF=FH=FB ; LAHB= 7, (106°, Cor. 4) and H is the foot of the altitude from B. Similarly, K and G are feet of the altitudes from C and A. Again, <KPH=LKFH=2_KAH. And A, K, O, H concyclic (107°), and AO is a diameter of the circumcircle, therefore P is the middle point of AO. Similarly, Q is the middle point of BO, and R of CO. Def.-The circle S passing through the nine points D, E, F, G, H, K, and P, Q, R, is called the nine-points circle of the AABC. Cor. Since the nine-points circle of ABC is the circumcircle of ADEF, whereof the sides are respectively equal to half the sides of the AABC, therefore the radius of the ninepoints circle of any triangle is one-half that of its circumcircle. are EXERCISES. 1. In 105° when P passes B where is the LAPB ? 2. A, B, C, D are four points on a circle whereof CD is a diameter and E is a point on this diameter. If ZAEB=2LACB, E is the centre. 3. The sum of the alternate angles of any octagon in a circle is six right angles. 4. The sum of the alternate angles of any concyclic polygon of 2n sides is 2(11 – 1) right angles. 5. If the angle of a trammel is 60° what arc of a circle will it describe? what if its angle is no ? 6. Trisect a right angle and thence show how to draw a regular 12-sided polygon in a circle. 7. If r, r' be the radii of two circles, and d the distance between them, the circles touch when d=r=r'. 8. Give the conditions under which two circles have 4, 3, 2, or i common tangent. 9. Prove Ex. 2, 116°, by drawing common tangents to the circles at P, Q, R, and S. 10. A variable chord passes through a fixed point on a circle, to find the locus of the middle point of the chord. II. A variable secant passes through a fixed point, to find the locus of the middle point of the chord determined by a fixed circle. 12. In Ex. II, what is the locus of the middle point of the secant between the fixed point and the circle? 13. In a quadrangle circumscribed to a circle the sums of the opposite sides are equal in pairs; and if the vertices be joined to the centre the sums of the opposite angles at the centre are equal in pairs. 14. If a hexagon circumscribe a circle the sum of three alternate sides is equal to that of the remaining three. 15. If two circles are concentric, any chord of the outer which is tangent to the inner is bisected by the point of contact; and the parts intercepted on any secant between the two circles are equal to one another. 16. If two circles touch one another, any line through the point of contact determines arcs which subtend equal angles in the two circles. 17. If any two lines be drawn through the point of contact of two touching circles, the lines determine arcs whose chords are parallel. 18. If two diameters of two touching circles are parallel, the transverse connectors of their end-points pass through the point of contact. 19. The shortest chord that can be drawn through a given point within a circle is perpendicular to the centre-line through that point. 20. Three circles touch each other externally at A, B, and C. The chords AB and AC of two of the circles meet the third circle in D and E. Prove that DE is a diameter of the third circle and parallel to the common centre line of the other two. 21. A line which makes equal angles with one pair of oppo site sides of a concyclic quadrangle makes equal angles with the other pair, and also with the diagonals. 22. Two circles touch one another in A and have a common tangent BC. Then LBAC is a right angle. 23. OA and OB are perpendicular to one another, and AB is variable in position but of constant length. Find the locus of the middle point of AB. 24. Two equal circles, touch one another and each touches one of a pair of perpendicular lines. What is the locus of the point of contact of the circles ? 25. Two lines through the common points of two intersecting circles determine on the circles arcs whose chords are parallel. 26. Two circles intersect in A and B, and through B a secant cuts the circles in C and D. Show that LCAD is constant, the direction of the secant being variable. 27. At any point in the circumcircle of a square one of the sides subtends an angle three times as great as that subtended by the opposite side. 28. The three medians of any triangle taken in both length and direction can form a triangle. SECTION VI. CONSTRUCTIVE GEOMETRY, INVOLVING THE PRINCIPLES OF THE FIRST FIVE SECTIONS, ETC. 117o. Constructive Geometry applies to the determination of geometric elements which shall have specified relations to given elements. Constructive Geometry is Practical when the determined elements are physical, and it is Theoretic when the elements are supposed to be taken at their limits, and to be geometric in character. (129) Practical Constructive Geometry, or simply “Practical Geometry," is largely used by mechanics, draughtsmen, surveyors, engineers, etc., and to assist them in their work numerous aids known as Mathematical Instruments " have been devised. A number of these will be referred to in the sequel. In “Practical Geometry” the “Rule” (16°) furnishes the means of constructing a line, and the “Compasses” (92°) of constructing a circle. In Theoretic Constructive Geometry we assume the ability to construct these two elements, and by means of these we are to determine the required elements. 118°. To test the “Rule.” Place the rule on a plane, as at R, and draw a line AB R A R B along its edge. Turn the rule into the position R'. If the edge now coincides with the line the rule is true. This test depends upon the property that two finite points A and B determine one line. (24°, Cor. 2) Def.-A construction proposed is in general called a proposition (2o) and in particular a problem. A complete problem consists of (1) the statement of what is to be done, (2) the construction, and (3) the proof that the construction furnishes the elements sought. B 119o. Problem.—To construct the right bisector of a given line segment. Let AB be the given segment. Construction.— With A and B as centres and with a radius AD greater than half of Ā AB describe circles. Since AB is < the sum of the radii and > their difference, the circles will meet in two points P and Q. (113°, Cor. 2, e) The line PQ is the right bisector required. Proof.-P and Q are each equidistant from A and B and ... they are on the right bisector of AB ; (54) .. PQ is the right bisector of AB. Cor. 1. The same construction determines C, the middle point of AB. Cor. 2. If C be a given point on a line, and we take A and B on the line so that CA=CB, then the right bisector of the segment AB passes through C and is I to the given line. is the construction gives the perpendicular to a given line at a given point in the line. 120°. Problem.—To draw a perpendicular to a given line from a point not on the line. |