College of education mathematics department
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- Điều hướng trang này:
- Exercise 3.5.
- Exercise 3.6.
- CHAPTER 4: GEOMETRIC ( TRIANGLE ) INEQUALITY
| Exercise 3.4. If are real numbers, prove that
When does equality hold? Solution: Let By Minkowski's inequality (for ) we have Hence the LHS of the desired inequality is not greater than , while the RHS is equal to . Now it is sufficient to prove that The last inequality can be reduced to the trivial . The equality in the initial inequality holds if and only if for some and Exercise 3.5. If are real number, prove that Solution: We will prove first by induction that: The case is a consequence of the Minkowski's inequality for and sequences . If the inequality holds for some , then The equality holds if and only if for some fixed real number and real numbers . Applying this to the sequences we get The equality holds if and only if . Exercise 3.6. Solve the equation Solution: Applying Minkowski's inequality, we have Equality holds if and only if . Hence the root of the given equation is . These inequalities in most cases have as variables the lengths of the sides of a given triangle; there are also inequalities in which appear other elements of the triangle, such as lengths of heights, lengths of medians, lengths of the bisectors, angles, etc. CHAPTER 4: GEOMETRIC ( TRIANGLE ) INEQUALITYFirst we will introduce some standard notation which will be used in this section: -lengths of the altitudes drawn to the sides , respectively. -lengths of the medians drawn to the sides , respectively. -lengths of the bisectors of the angles , respectively. -area, -semi-perimeter, - circumradius, -inradius. Furthermore we will give relations between the lengths of medians and lengths of the bisectors of the angles with the sides of a given triangle. Namely we have and We can rewrite the last three identities in the following form Also we note that the following properties are true, and we'll present them without proof. (The first inequality follows by using geometric formulas and mean inequalities, and the second inequality immediately follows, for instance, according to Leibniz's theorem.) tải về 344.73 Kb. Chia sẻ với bạn bè của bạn:
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