B A C D E F

P/s: Hình vẽ chỉ mang tính chất minh họa

Từ D kẻ các đường song song với AC,AB cắt AB,AC lần lượt tại E,F

=> Tứ giác AEDF là hình bình hành

Lại có AD là phân giác góc EAF => Tứ giác AEDF là hình thoi

=> AE = ED = DF = FA

Xét trong tam giác AED cân tại E có góc EAD = 60 độ

=> Tam giác AED đều => AD = DE = DF

Áp dụng định lý Thales ta có: 

DE // AC => \(\frac{DE}{AC}=\frac{BD}{BC}\)      ;  DF // AB => \(\frac{DF}{AB}=\frac{DC}{BC}\)

Cộng vế với vế 2 đẳng trên ta được: \(\frac{DE}{AC}+\frac{DF}{AB}=\frac{BD}{BC}+\frac{DC}{BC}\)

\(\Leftrightarrow\frac{AD}{AC}+\frac{AD}{AB}=1\Rightarrow\frac{1}{AB}+\frac{1}{AC}=\frac{1}{AD}\)

=> đpcm

a)      \(2KMnO_4\underrightarrow{t^0}K_2MnO_4+MnO_2+O_2\)

b)     Đặt x là số mol của \(KMnO_4\)

        \(\Rightarrow n_{K_2MnO_4}=\frac{1}{2}n_{KMnO_4}=\frac{1}{2}x\)

Ta có : \(m_{KMnO_4}-m_{K_2MnO_4}=2,4\left(g\right)\)

\(\Leftrightarrow158x-\frac{197}{2}x=2,4\)

\(\Leftrightarrow x\approx0,04\left(mol\right)\)

\(\Rightarrow m_{KMnO_4}=0,04\times158=6,32\left(g\right)\)

hãy viết một đoạn văn 100 chữ phải bắt đầu bằng câu "I was standing beside someone famous"

I was standing beside someone famous! I was so excited that during a few seconds I couldn't do anything. I was standing and looked him. He was more taller and more beautiful that he seemed on television. Suddenly he turned his head and fixed me. I was petrified! He smile and said hello. I opens my mouth but I couldn't say something. He asked me how I found the show. Finally I answered him. I said it was great. I added that I lived his music and parular his last album. I asked him to sign me an autograph the she shakes my hand and went.

Đề bài thiếu hở :??

cần gấp

\(\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+...+\frac{1}{\left(2n-1\right)\left(2n+1\right)}\)

\(=\frac{1}{2}\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{\left(2n-1\right)\left(2n+1\right)}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2n-1}-\frac{1}{2n+1}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{2n+1}\right)\)

\(=\frac{1}{2}\cdot\frac{2n+1-1}{2n+1}\)

\(=\frac{1}{2}\cdot\frac{2n}{2n+1}=\frac{n}{2n+1}\)