I don't know if there are people who recognize the following pattern in Isosceles triangles and they are new to me, it's difficult to understand and I've been working on it and doesn't seem to get nowhere. Any explanation?
Table of isosceles triangles.
$$ \begin{array}{1|2|3|4|5|6} \\ \hline 1& 1+1+1=3 & 2+2+1=5 & 3+3+1=7 & 4+4+1=9 & 5+5+1=11\\ 3 & & 2+2+2=6 & 3+3+2=8 & 4+4+2=10 & 5+5+2=12\\ 5 & & 2+2+3=7 & 3+3+3=9 & 4+4+3=11 & 5+5+3=13 \\ 7 & & & 3+3+4=10 & 4+4+4=12 & 5+5+4=14 \\ 9 & & & 3+3+5=11 & 4+4+5=13 & 5+5+5=15 \\ 11& & & & 4+4+6=14 & 5+5+6=16 \\ 13& & & & 4+4+7=15 & 5+5+7=17 \\ 15 & & & & & 5+5+8=18 \\ 17 & & & & & 5+5+9=19 \\ \hline n & 3 & 18=3+3\times 5 & 18+3\times 9=45 & 45+3\times 13=84 & 84+3\times 17=135 \end{array}$$
$a+b>c$ $a+c>b$ $b+c>a$
And the law of the triangles says there can only be 9 triangles for the fifth rank
$$ \begin{array}{1|2|3|4|5} \\ \hline 5th rank & \cos A & \ cos B & \cos C & \cos A +\cos B+\cos C \\ \hline 5+5+1=11 & \frac {1}{10} & \frac {1}{10} & \frac {50-1^2}{50} & \frac {59}{50}\\ \hline 5+5+2=12 & \frac {2}{10} & \frac {2}{10} & \frac {50-2^2}{50} & \frac {66}{50}\\ \hline 5+5+3=13 & \frac {3}{10} & \frac {3}{10} & \frac {50-3^2}{50} & \frac {71}{50}\\ \hline 5+5+4=14 & \frac {4}{10} & \frac {4}{10} & \frac {50-4^2}{50} & \frac {74}{50}\\ \hline 5+5+5=15 & \frac {5}{10} & \frac {5}{10} & \frac {50-5^2}{50} & \frac {75}{50}\\ \hline 5+5+6=16 & \frac {6}{10} & \frac {6}{10} & \frac {50-6^2}{50} & \frac {74}{50}\\ \hline 5+5+7=17 & \frac {7}{10} & \frac {7}{10} & \frac {50-7^2}{50} & \frac {71}{50}\\ \hline 5+5+8=18 & \frac {8}{10} & \frac {8}{10} & \frac {50-8^2}{50} & \frac {66}{50}\\ \hline 5+5+9=19 & \frac {9}{10} & \frac {9}{10} & \frac {50-9^2}{50} & \frac {59}{50}\\ \hline \end{array} $$
$1\times 1\times 2=2$
$2\times 2\times 2=8$
$3\times 3\times 2=18$
$4\times 4\times 2=32$
$5\times 5\times 2=50$