PROPOSITION XL. PROBLEM. M Let A B C be the given triangle. Draw the line A E, bisecting 2 A, Draw EHI to the line A C. The O K H M is the O required. and EM I to BC. Iden. Cons. .. A A KE=A A HE, § 110 (Two rt. A are equal if the hypotenuse and an acute Z of the one be equal respectively to the hypotenuse and an acute 2 of the other). i. EK= EH,, (being homologous sides of equal A). . EK, EH, and E M are all equal. sa o described from E as a centre, with a radius equal to E H, will touch the sides of the A at points H, K, and M, and be inscribed in the A. Q. E. F. § 174 PROPOSITION XLI. PROBLEM. 243. Upon a given straight line, to describe a segment which shall contain a given angle. H Let A B be the given line, and M the given angle. It is required to describe a segment upon the line A B, which shall contain 2 M. At the point B construct Z A B E equal to Z M. Bisect the line A B by the | FH. From the point B, draw BO I to È B. From 0, the point of intersection of FH and B O, as a centre, with a radius equal to 0 B, describe a circumference. Now the point 0, being in a I erected at the middle of A B, is at equal distances from A and B, $ 58 (every point in a I erected at the middle of a straight line is at equal dis tances from the extremities of that line); .. the circumference will pass through A. Cons. ..BE is tangent to the O, $ 186 (a straight line I to a radius at its extremity is tangent to the O). ..LABE is measured by 1 arc AB, $ 209 (being an Z formed by a tangent and a chord). Also any Z inscribed in the segment A H B, as for instance Ć A KB, is measured by 1 arc A B, § 203 (being an inscribed Z). ..LA KB = L A B E, .. Z A K B = M. Q. E. F. PROPOSITION XLII. PROBLEM. 244. To find the ratio of two commensurable straight lines. E H LIB CL Let A B and C D be two straight lines. It is required to find the greatest common measure of A B and C D, so as to express their ratio in figures. Apply C D to A B as many times as possible. Suppose twice with a remainder E B. Suppose three times with a remainder F D. Suppose once with a remainder H B. Suppose once with a remainder K D. Suppose K D is contained just twice in H B. The measure of each line, referred to KD as a unit, will then be as follows: HB = 2 KD; . AB 41 KD A B 41 .. the ratio of CD=18 Q. E.' EXERCISES. 1. If the sides of a pentagon, no two sides of which are parallel, be produced till they meet; show that the sum of all the angles at their points of intersection will be equal to two right angles. 2. Show that two chords which are equally distant from the centre of a circle are equal to each other; and of two chords, that which is nearer the centre is greater than the one more remote. 3. If through the vertices of an isosceles triangle which has each of the angles at the base double of the third angle, and is inscribed in a circle, straight lines be drawn touching the circle ; show that an isosceles triangle will be formed which has each of the angles at the base one-third of the angle at the vertex. 4. A D B is a semicircle of which the centre is C'; and A EC is another semicircle on the diameter AC; AT is a common tangent to the two semicircles at the point A. Show that if from any point F, in the circumference of the first, a straight line FC be drawn to C, the part FK, cut off by the second semicircle, is equal to the perpendicular FH to the tangent A T. 5. Show that the bisectors of the angles contained by the opposite sides (produced) of an inscribed quadrilateral intersect at right angles. 6. If a triangle A B C be formed by the intersection of three tangents to a circumference whose centre is 0, two of which, A M and A N, are fixed, while the third, BC, touches the circumference at a variable point P; show that the perimeter of the triangle A B C is constant, and equal to AM + AN, or 2 A M. Also show that the angle B O C is constant. 7. A B is any chord and AC is tangent to a circle at A, C D E a line cutting the circumference in D and E and parallel to A B; show that the triangle A C D is equiangular to the triangle E A B. CONSTRUCTIONS. 1. Draw two concentric circles, such that the chords of the outer circle which touch the inner may be equal to the diameter of the inner circle. 2. Given the base of a triangle, the vertical angle, and the length of the line drawn from the vertex to the middle point of the base : construct the triangle. 3. Given a side of a triangle, its vertical angle, and the radius of the circumscribing circle : construct the triangle. 4. Given the base, vertical angle, and the perpendicular from the extremity of the base to the opposite side : construct the triangle. 5. Describe a circle cutting the sides of a given square, so that its circumference may be divided at the points of intersection into eight equal arcs. 6. Construct an angle of 60°, one of 30°, one of 120°, one of 150°, one of 45°, and one of 135o. 7. In a given triangle A B C, draw Q D E parallel to the base B C and meeting the sides of the triangle at D and E, so that D E shall be equal to DB + EC. 8. Given two perpendiculars, A B and C D, intersecting in 0, and a straight line intersecting these perpendiculars in E and F; to construct a square, one of whose angles shall coincide with one of the right angles at 0, and the vertex of the opposite angle of the square shall lie in E F. (Two solutions.) 9. In a given rhombus to inscribe a square. 10. If the base and vertical angle of a triangle be given ; find the locus of the vertex. 11. If a ladder, whose foot rests on a horizontal plane and top against a vertical wall, slip down; find the locus of its middle point. |