Theo t/c dãy tỉ số bằng nhau ta có :
\(\dfrac{ab+ac}{2}=\dfrac{bc+ba}{3}=\dfrac{ca+cb}{4}\)
\(=\dfrac{ab+ac+bc+ba-ca-cb}{2+3-4}=\dfrac{2ab}{1}\) \(\left(1\right)\)
\(=\dfrac{bc+cb+bc+ba-ab-ac}{3+4-2}=\dfrac{2bc}{5}\left(2\right)\)
\(=\dfrac{ab+ac+ca+cb-bc-ba}{2+4-3}=\dfrac{2ac}{3}\)\(\left(3\right)\)
Từ \(\left(1\right)+\left(2\right)+\left(3\right)\Leftrightarrow\dfrac{2ab}{1}=\dfrac{2bc}{5}=\dfrac{2ac}{3}\)
\(\dfrac{2ab}{1}=\dfrac{2bc}{5}\Leftrightarrow\dfrac{a}{1}=\dfrac{c}{15}\) \(\Leftrightarrow\dfrac{a}{3}=\dfrac{c}{15}\left(I\right)\)
\(\dfrac{2bc}{5}=\dfrac{2ac}{3}\Leftrightarrow\dfrac{b}{5}=\dfrac{a}{3}\left(II\right)\)
Từ \(\left(I\right)+\left(II\right)\Leftrightarrow\dfrac{a}{3}=\dfrac{b}{5}=\dfrac{c}{15}\left(đpcm\right)\)