$\textbf{Cauchy-Goursat Theorem}:$ If a function $f(z)$ is analytic within and on a closed contour $C$, then $\oint_{C}f(z)dz=0.$
In many proofs they start from a triangle
$\bigtriangleup_{0}=\bigtriangleup(a,b,c)=\{\mu a+\lambda b+\gamma c\:|\:\mu+\lambda+\gamma=1\text{ and } \mu,\lambda,\gamma\geq0\}\subseteq C$
and then subdivide $\bigtriangleup_{0}$ into four smaller triangles
$\bigtriangleup_{1}=\bigtriangleup(a,b',c'),\bigtriangleup_{2}=\bigtriangleup(a',b',c),\bigtriangleup_{3}=\bigtriangleup(a',b',c'),\bigtriangleup_{4}=\bigtriangleup(a',b,c')$ taking the points $a'=\frac{(b+c)}{2},b'=\frac{(a+c)}{2}$ and $c'=\frac{(a+b)}{2}$.
Now, they assume that $d(\bigtriangleup_{i})=\frac{d(\bigtriangleup_{0})}{2}$ and $p(\bigtriangleup_{i})=\frac{p(\bigtriangleup_{0})}{2}$ where $i=1,2,3,4$ and $p,d$ represents the perimeter and the diameter of each triangle.
Could you give me a rigorous proof about that property? Thank you so much.
Each of the smaller triangles are similar to the larger one. This should be fairly obvious from drawing a picture, but you can prove it just with simple arithmetic (write out what the side lengths actually are).