4. If two triangles be upon the same base, but the vertex of one within the other triangle, then the perimeter of the inner triangle is less than that of the other. [Perimeter means sum of the sides.] 5. Two sides of a triangle being supposed given in magnitude, show by any method that the greater the angle between them, the greater the third side. 6. If one angle of a triangle be equal to the sum of the other two, the triangle can be divided into two isosceles triangles. XII. 1. ABC is a triangle, and the angle A is bisected by a straight line which meets BC in D. Show that BA is greater than BD, and CA greater than CD. 2. If a straight line be drawn through A, one of the angular points of a square, cutting one of the opposite sides, and meeting the other produced in F, show that AF is greater than the diagonal of the square. 3. The two sides of any triangle are together greater than twice the straight line drawn from the vextex to the middle of the base. 4. Two equal circles have an angle at the centre of each ; if these are unequal, the chord opposite the greater angle is greater than that opposite the other. 5. In two equal circles, having a chord in one greater than a chord in the other, show that the angle subtended at the centre of its circle by the first chord is greater than that similarly subtended by the second. 6. Show that three rods pivoted together at their extremities will form a rigid triangle ; that is, no two sides will work about the pivot at their common extremity. XIII. 1. From the vertex A of an isosceles triangle, two straight lines are drawn, one to meet the base BC in D, and the other to meet BC produced in E; show that AB is less than AE, and greater than AD. 2. The sum of the distances of any point from the three angles of a triangle is greater than half the sum of the sides. 3. If the vertical angle of a triangle be equal to the sum of the other two, the straight line drawn from the vertex to the middle of the base will be equal to half the base. [See XI. 6.] 4. ABCD is a quadilateral, of which AD is the longest side and BC the shortest ; show that the angle ABC is greater than the angle ADC, and the angle BCD greater than the angle BAD. (Join BD to begin with.) 5. Prove Euclid I. 18, given AC greater than AB in the triangle ABC, by cutting AD off AC equal to AB, and joining D to E, where E is the point in which the bisector of the angle A meets 'BC. 6. I have a quadrilateral framework of four rods jointed at their extremities; prove that it will become rigid if I fasten to two opposite corners of it the two ends of another rigid rod. XIV. 1. If two chords be drawn from one point in the circumference of a circle, one of them passing through the centre, this latter will be longer than the other. 2. The four sides of a quadrilateral are together greater than the sum of its diagonals. 3. If straight lines be drawn from a point within a triangle to each angle, the sum of them shall be less than the perimeter of the triangle. 4. Construct a triangle, given the base, an angle at the base, and the sum of the two sides. 5. Show that, of the two diagonals of a parallelogram which is not a rectangle, the greater connects the two acute, and the lesser the two obtuse angles of the figure. 6. Of all the straight lines which can be drawn from a given point to a given straight line (a) The perpendicular is less than any other; also equal. also less than it. From (6) and (c) infer that not more than two of the lines can be equal to one another. SECTION V.-PROPOSITION XXVI. XV. 1. Prove that a triangle is isosceles if the bisector of any angle is perpendicular to the opposite side. 2. If a straight line be drawn from the vertical angle of an isosceles triangle perpendicular to the base, it shall bisect the base. 3. The same straight line shall bisect the vertical angle. 4. If from any point in the same perpendicular two straight lines be drawn to the ends of the base, they will form with the base an isosceles triangle. 5. If a straight line be drawn bisecting an angle, any point in it is equidistant from the two straight lines which contain the angle. 6. A straight line terminated by two parallel straight lines is bisected at a point C. Prove that every straight line through Cwhich is terminated by the parallels is bisected at C. XVI. 1. Given an angle BAC and a point D; show that a perpendicular from D upon the bisector of the angle A will cut off equal distances along AB, AC. 2. Through a given point draw a straight line which shall be equally inclined to two given straight lines. 3. Find a point in the base of any given triangle which is equidistant from the sides. 4. Two right-angled triangles are equal in all respects if the hypotenuse and an acute angle of one are respectively equal to similar parts of the other. 5. From a given point draw a straight line which shall be equidistant from two other given points. 6. From any point in an isosceles triangle, which not in the bisector of its vertical angle, draw straight lines to the ends of the base; and show that these are not equal. Is the new triangle thus formed necessarily scalene ? SECTION VI. PROPOSITIONS XXVII.-XXXI. XVII. 1. Through a given point on either side of a given indefinite right line draw the perpendicular and the parallel to the line. 2. A parallel to the base BC of a triangle ABC cuts the sides, produced if necessary, at D and E. Prove that the new triangle ADE has its angles equal to those of the original triangle ABC. 3. Show that every parallel to the base of an isosceles triangle cuts off equal lengths on the sides measured from the vertex of the triangle. 4. If a straight line be drawn parallel to one side of a parallelogram through an internal point, it divides the figure into two parallelograms. 5. Two rectilinear segments of any lengths being supposed to have a common middle point, but not a common direction, show that they are the two diagonals of a parallelogram. 6. Show how to construct a triangle having its vertex at a given point, its base on a given indefinite right line, and its base angles equal to two given angles. XVIII. 1. Let ABC be any triangle; at A, and on the side of AB remote from C, make an angle BAE equal to ABC; then prove that a parallel has been drawn to BC. 2. In the triangle ABC, having AB produced to D, make at B, on the same side of DB as C, an angle DBF equal to BAC; then BF will be parallel to CA. 3. In the triangle ABC, make at C, on the same side of BC as A, an angle BCH equal to the supplement of ABC; then CH will be parallel to AB. 4. If a quadrilateral have two adjacent angles equal to two right angles, it will also have two sides parallel. 5. If a straight line be parallel to the base of an isosceles triangle, it is equally inclined to the two sides or the two sides produced. 6. If a parallelogram be formed out of bars pivoted together at the angles, when one angle is opened out, show that one diagonal of the figure is increased and the other diminished. XIX. 1. Two intersecting straight lines which are terminated in two parallel straight lines shall bisect one another, if they cut off equal portions along those parallels. 2. If a straight line be terminated by two parallels, any other straight line passing through its middle point will be bisected there, if its ends be in the parallels. 3. If two straight lines be respectively parallel to two others, the angle between the first two will be equal to the angle between the other two. 4. If the straight line which bisects the exterior angle at the vertex of a triangle be parallel to the base, the triangle is isosceles. 5. If an angle be bisected, the two straight lines drawn from any point in the bisector, each parallel to one side of the angle and meeting the other, will be equal. 6. A straight line drawn at right angles to the base of an isosceles triangle cuts one side in P, the other produced in Q. If R be the vertex of the original triangle, show that the triangle PQR is isosceles also. SECTION VII. PROPOSITION XXXII. AND COROLLARIES. XX. 1. A triangle is obtuse-angled if one angle be greater than the sum of the other two, right-angled if equal, and acute-angled ifeach angle be less. 2. A straight line which makes equal angles with two sides of an isosceles triangle will be parallel to the base. 3. In an isosceles triangle, each angle at the base is equal to half the exterior angle at the vertex. 4. If two triangles have the sum of two angles in one trian gle equal to the sum of two angles in the other, their third angles are also equal. 5. Each angle of an equilateral triangle is two-thirds of one right angle. Apply this in trisecting a right angle. 6. Determine an exterior and an interior angle of a regular pentagon. XXI. 1. A triangle is right-angled if the straight line drawn from the vertex to the middle point of the base be equal to half the base. . 2. The vertical angle of the same triangle is acute if this line be greater than half the base; but obtuse if less. 3. If the sides of a regular pentagon be produced to meet in five points, the angles at these points are together equal to two right angles. If the figure were a hexagon, the angles would be equal to four right angles. 4. The straight line joining the middle point of the hypotenuse of a right-angled triangle to the right angle, is equal to half the hypotenuse. 5. Construct a right-angled triangle, having given the hypotenuse and the sum of the sides. 6. Construct a right-angled triangle, having given the hypotenuse and the difference of the sides. XXII. 1. Prove that no polygon of the ordinary form can have more than three of its exterior angles obtuse, or more than three of its interior angles acute. 2. Construct an equilateral triangle having its altitude given. 3. Construct an isosceles triangle whose base and angle at the vertex are given. 4. Two of the angles in a triangle, which is not given, are known to be equal to two given acute angles. Show how to make, with a given straight line, an angle which shall be equal to the third angle in the triangle. 5. Show how to construct a triangle, having given two angles and the side opposite one of them. 6. If one diagonal of a quadrilateral be diminished without altering the lengths of the sides, the other diagonal will be increased. |