College of education mathematics department
tải về 344.73 Kb.
|
- Điều hướng trang này:
- 4.2. Exercises
4.1. Proposition:For an arbitrary triangle the following inequalities hold Basic inequalities which concern the lengths of the sides of a given triangle are well-known inequalities: . But also useful and frequent substitutions are: The question is whether there are always positive real numbers , such that the above identities (3.1) hold and are the sides of the triangle. The answer is positive. Namely are tangent segments dropped from the vertices to the inscribed circle of the given triangle. From (3.1) we easily get that and then clearly . Remark: The substitutions (3.1) are called Ravi's substitutions. 4.2. Exercises:Exercise 4.1. Let be the lengths of the sides of given triangle. Prove the inequalities Solution: Let's prove the right-hand inequality. Since we have , i.e. Similarly we get and . Therefore Let's consider the left-hand inequality. If we denote then we have Hence i.e. , as required. Remark The left-hand inequality is known as Nesbitt's inequality, and is true for any positive real numbers and . Exercise 4.2. Let be a triangle with side lengths and with side lengths . Prove that , where is the area of , and is the area of . Solution: By Heron's formula for and we have and Since and are the side lengths of triangle there exist positive real numbers such that . Now we easily get that So it suffices to show that Applying we obtain By (3.4) and (3.5) we get the desired result. Exercise 4.3. Let be the side lengths, and be the respective angles (in radians) of a given triangle. Prove the inequalities Solution: First let's prove the left inequality. We can assume that and then clearly . So we have i.e. Hence, Equality occurs if and only if . Let's consider the right inequality. Since and are side lengths of a triangle we have and . If we multiply these inequalities by and , respectively, we obtain i.e. tải về 344.73 Kb. Chia sẻ với bạn bè của bạn:
|