and let AE be any other line drawn from C to a point E in AB; it is required to prove that CD is less than CE. The exterior angle CDB of the triangle CED is greater than CED, an interior opposite angle. [Prop. 16] But the right angle CDE is equal to the right angle CDB; therefore the angle CDE is also greater than CED. Wherefore, by Prop. 19, the side CE of a triangle CED, which is opposite the angle CDE, is greater than the side CD which is opposite the angle CED. Q. E. D. 98. DEF. The distance between two points is the length of the straight line joining them. The distance of a point from a straight line is the length of the straight line drawn from the point perpendicular to that line. EXAMPLES XXIV. 1. The hypotenuse is the greatest side in a right-angled triangle. 2. D is the point in which the bisector of the angle BAC of the triangle ABC cuts the side BC; prove that the side AC is greater than CD and that the side AB is greater than BD. 3. In an isosceles triangle ABC in which AB= AC, D is any point in the line BC; prove that AB is greater than AD when D is between B and C and less than AD when D is not between B and C. 4. Of all straight lines which can be drawn from a given point to a straight line the perpendicular is the least; and of the others those which are nearer the perpendicular are less than those which are more remote. 5. In any triangle ABC, D is a point in BC; prove that AD is less than the greater of the two sides AB, AC. 6. In the triangle ABC, AC is greater than AB and D is taken in AC so that AD is equal to AB; prove that DC is less than BC. Proposition 20. 99. Any two sides of a triangle are together greater than the third side. Let ABC represent a triangle; it is required to prove that any two BA, AC of the sides are together greater than the third side BC. B Let D be the point on the side BA produced, join DC. Then, because AC is equal to AD, therefore the angle ACD is equal to ADC. [Prop. 5.] But the angle BCD is greater than its part ACD; therefore also the angle BCD is greater than ADC. Consider the triangle BCD; because the angle BCD is greater than BDC, therefore the side BD is greater than BC. [Prop. 19.] But BD, that is BA and AD, is equal to BA and AC together; therefore BA and AC together are greater than BC. Similarly it may be proved that any other two sides together are greater than the third side. Wherefore, any two sides of a triangle are together, etc. Example. Given a straight line AB and two points C, D, find the point P in the straight line such that the sum of PC, PD, its distances from the two given points, is the least possible. First let the points be on opposite sides of the line, then the point in AB which is required is found by joining DC which cuts AB in the point P. For if E be any other point in AB, then, by Prop. 20, CE and ED together are greater than the straight line CD. Next let the points be on the same side of the line. [Analysis. If we can find a point C' on the other side of AB, such that the distance of any point E in AB from C' is equal to the distance of E from C, we could find P by joining C'D.] Draw CH perpendicular to AB; and produce CH to C' so that C'H = HC. Then the distance of any point E from Cits distance from C', so that CE+ED=C'E+ED. And CE+ ED is least when C'E+ED is least, that is when C'D is a straight line. Wherefore P is on the straight line C'D. 100. By the aid of Proposition 20 it can be shewn that the length of the straight line joining two points is less than the length of any other line joining the two points. For suppose the line consists of a series of finite straight lines then it can be proved by a series of applications of Prop. 20 that the length of the straight line AB is less than the sum of a series of straight lines joining AB. And when the line is curved it may be considered to be made up of a multitude of very short finite straight lines. Hence, the straight line joining them, is the shortest distance between two points. EXAMPLES XXV. 1. Any three sides of a quadrilateral are together greater than the fourth side. 2. The straight line joining two points is less than any system of straight lines joining the two points. 3. The sum of the sides of a quadrilateral is greater than twice either of the diagonals. 4. The sum of the sides of a quadrilateral is greater than the sum of the diagonals. 5. The difference between any two sides of a triangle is less than the third side. 6. Given two points C, D and a straight line AB, find P in AB such that the difference between PC and PD is as long as possible. 7. Any two sides of a triangle together are greater than twice the line drawn from the middle point of the base (the third side) to the opposite angle. 8. Twice the sum of the distances of the angular points of a triangle from any point in its plane is greater than the perimeter of the triangle. [Perimeter=sum of the sides, i.e. all the way round.] 9. A man is riding from one town to another on the same side of a straight river; he wants to take his horse to drink at the river on the way; how can he find the shortest path to take in order to do this? 10. Prove Proposition 20 by bisecting one of the angles. [See XXI. 10.] |