Chapter 4. The angle bisectors. 4.1 The angle bisector theorem

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1 hapter 4 The angle bisetors 4.1 The angle bisetor theorem Theorem 4.1 (ngle bisetor theorem). The bisetors of an angle of a triangle divide its opposite side in the ratio of the remaining sides. If and respetively the internal and external bisetors of angle, then : = : b and : = : b. b Proof. onstrut lines through parallel to the bisetors and to interset the line at and. (1) Note that = = =. This means =. learly, : = : = : = : b. (2) Similarly, =, and : = : = : = : b.

2 140 The angle bisetors Exerise 1. Given a segment, eret a square on it, and an adjaent one with base. If D is the vertex above, onstrut the bisetor of angle D to interset at P. alulate the ratio P : P. D P 2. square is insribed in a right triangle with sidesaandb. Show that eah side of the square has length l = ab a+b. a l l b 3. In the diagram, D, D, Q, and P are squares. Show that,p,, Q are onyli. Q D D P

3 4.1 The angle bisetor theorem Given a right triangle with a right angle at and α β, let P be the intersetion of the median on and the bisetor of angle. Show that lines throughp perpendiular to the sides of the triangle ut out triangles of equal areas. D F P 5. Let be a triangle with = 2. The internal bisetor of angle meets at D. Let M and N be the midpoints of and D respetively. Suppose that, M, D,N are onyli. Prove that = In triangle, α = 120. is the bisetor of angle. Show that 1 t = 1 b + 1. t b 7. In the diagram below,,, and D are equilateral triangles. Suppose = 120. Show that 1 b = 1 a a b D 1 Hint: Extend to interset at T. Show that T = a.

4 142 The angle bisetors 4.2 Steiner-Lehmus theorem Theorem 4.2. triangle is isoseles if it has two equal angle bisetors. F E Proof. Suppose the bisetorse = F, but triangle not isoseles. We may assume <. onstrut parallels to through E and F to interset and at and respetively. (1) In the isoseles trianglese andf with equal basese andf, E < F = E < F. (2) = F = < = E. Therefore, F E = + 1 < E + 1 =, and > E. This learly implies F < E, E E ontraditing (1) above. Exerise 1. is a triangle with both external angle bisetors t b and t equal to a. alulate the angles of the triangle. t b a t 2. The bisetor t a and the external bisetor t b of triangle satisfy t a = t b =. alulate the angles of the triangle.

5 4.2 Steiner-Lehmus theorem 143 t b t a The lengths of the bisetors Proposition 4.1. (a) The lengths of the internal and external bisetors of angle are respetively t a = 2b b+ os α 2 and t a = 2b b sin α 2. t a b t a Proof. Let and be the bisetors of angle. (1) onsider the area of triangle as the sum of those of triangles and. We have 1 2 t a(b+)sin α 2 = 1 2 bsinα. From this, t a = b b+ sinα sin α 2 = 2b b+ os α 2. (2) onsider the area of triangle as the differene between those of and. Remarks. (1) 2b b+ is the harmoni mean of b and. It an be onstruted as follows. If the perpendiular to at intersets and at and, then = = 2b b+.

6 144 The angle bisetors t a b (2) pplying Stewart s Theorem with λ = and µ = ±b, we also obtain the following expressions for the lengths of the angle bisetors: ( ( ) ) 2 a t 2 a = b 1, b+ ( ( ) 2 t 2 a a = b 1). b Exerise 1. The lengths of the sides of a triangle are 84, 125, 169. alulate the lengths of its internal bisetors is a right triangle in whih the bisetor of the right angle, and the median to the hypotenuse have lengths 24 2 and 35 respetively. alulate the sidelengths of the triangle. 3. (a) In triangle,a = 5,b = 8, = 7. Show thatt a : t b = b : a. 5 8 t a t b 7 (b) Suppose t a : t b = b : a. Show that the triangle is either isoseles, orγ = nswers: 975 7, ,

7 4.2 Steiner-Lehmus theorem (a) In triangle, a = 7, b = 5 and = 3. Let,, be the angle bisetors. Show that is a right triangle. (b) If,, are the angle bisetors of triangle, show that = 2a2 b(b 2 +b+ 2 a 2 ) (b+) 2 (+a)(a+b). I () Given a segment, onstrut the lous of for whih = Find a right triangle for whih the bisetor of an aute angle is the geometri mean of the two segments it divides on the opposite side. a t E 6. Find an isoseles triangle for whih the bisetor of a base angle is the geometri mean of the two segments it divides on the opposite side. 3 3 nswer: (a,b,) = (1, 1+ 2, 1+ 2).

8 146 The angle bisetors 7. In,α = 60, andβ < γ. The bisetor of intersets at. If is a mean proportional between and, findβ. 8. The bisetor of angle of triangle intersets at. Show that is the geometri mean of and if and only if b+ = 2a. 9. Let t a, t b, t be the lengths of the bisetors of a triangle, and T a, T b, T these angle bisetors extended until they are hords of the irumirle. Prove that ab = t a t b t T a T b T. I

9 4.3 The irle of pollonius The irle of pollonius Theorem 4.3. and are two fixed points. For a given positive numberk 1, 4 the lous of points P satisfying P : P = k : 1 is the irle with diameter, where and are points on the line suh that : = k : 1 and : = k : 1. P O Proof. Sine k 1, points and an be found on the line satisfying the above onditions. onsider a point P not on the line with P : P = k : 1. Note that P and P are respetively the internal and external bisetors of angle P. This means that angle P is a right angle, andp lies on the irle with as diameter. onversely, let P be a point on this irle. We show that P : P = k : 1. Let be a point on the line suh that P bisets angle P. Sine P and P are perpendiular to eah other, the line P is the external bisetor of angle P, and = = =. On the other hand, = = =. omparison of the two expressions shows that oinides with, andp is the bisetor of angle P. It follows that P = = k. P 4 Ifk = 1, the lous is learly the perpendiular bisetor of the segment.

10 148 The angle bisetors Exerise 1. If = d, and k 1, the radius of the pollonius irle is k k 2 1 d. 2. Suppose is a triangle with, and letd,e,, be points on the line defined as follows: D is the midpoint of, E is the foot of the perpendiular from to,, biset angle. Prove that = DE. E D 3. Given two disjoint irles () and (), find the lous of the point P suh that the angle between the pair of tangents fromp to() and that between the pair of tangents from P to () are equal. P

Chapter 3. The angle bisectors. 3.1 The angle bisector theorem

Chapter 3. The angle bisectors. 3.1 The angle bisector theorem hapter 3 The angle bisectors 3.1 The angle bisector theorem Theorem 3.1 (ngle bisector theorem). The bisectors of an angle of a triangle divide its opposite side in the ratio of the remaining sides. If