I was said that a geometrical method is very useful to solve problems. Do you have any idea? Any book can you suggest me to read?

Thanks.

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- Apr 3rd 2011, 03:48 AMLuyenTDHGeometrical method
I was said that a geometrical method is very useful to solve problems. Do you have any idea? Any book can you suggest me to read?

Thanks. - Apr 3rd 2011, 04:18 AMveileen
Well, if "geometrical" includes trigonometry and vectors I have some problems that can be solved easier like that. Also geometrical method is useful in problems with complex numbers (and, yes, I have problems that can be solved easier that way too).

Books - um, a good and complet one that I know is "Complex numbers from A to Z" - Titu Andreescu, Dorin Andrica. - Apr 3rd 2011, 04:33 AMLuyenTDHGeometrical methods
Thanks for Veileen's soon comments.

By chance, do you know about the Titu's lemma? - Apr 3rd 2011, 04:39 AMveileen
"Titu's lemma" Do you refer at that lemma that can be deduced from Cauchy's inequality? If so, yes, I know ^^

- Apr 3rd 2011, 04:51 AMLuyenTDHTitu's lemma
- Apr 3rd 2011, 05:06 AMveileen
$\displaystyle \frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^ 2}{b_n}\geq \frac{(a_1+a_2+...+a_n)^2}{b_1+b_+...+b_n},$

Equality if and only if $\displaystyle \frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n }$

Demonstration:

$\displaystyle (x_1^2+x_2^2+...+x_n^2)(y_1^2+y_2^2+...+y_n^2)\geq (x_1y_1+x_2y_2+...+x_ny_n)^2$ (Cauchy's inequality)

Let $\displaystyle x_i=\frac{a_i}{\sqrt{b_i}}$ and $\displaystyle y_i=\sqrt{b_i}$, then:

$\displaystyle \sum_{i=1}^{n}\frac{a_i^2}{{b_i}}\cdot \sum_{i=1}^{n}b_i\geq \left ( \sum_{i=1}^{n}a_i^2 \right )\Rightarrow$ $\displaystyle \sum_{i=1}^{n}\frac{a_i^2}{{b_i}}\geq \frac{\left ( \sum_{i=1}^{n}a_i^2 \right )}{\sum_{i=1}^{n}b_i}$ which is exactly Titu's inequality.