# complex analysis – Questions about triangles related to the Cauchy-Goursat theorem

$$textbf {Cauchy-Goursat theorem}:$$ If a function $$f (z)$$ is analytic inside and on a closed contour $$C$$then $$oint_ {C} f (z) dz = 0.$$

In a lot of evidence, they start from a triangle

$$bigtriangleup_ {0} = bigtriangleup (a, b, c) = { mu a + lambda b + gamma c : | : mu + lambda + gamma = 1 text {and} mu, lambda, gamma geq0 } subseteq C$$

then subdivide $$bigtriangleup_ {0}$$ in four small triangles

$$bigtriangleup_ {1} = bigtriangleup (a, b, c), bigtriangleup_ {2} = bigtriangleup (a, b, c), bigtriangleup_ { 3} = bigtriangleup (a, b, c), bigtriangleup_ {4} = bigtriangleup (a, b, c)$$ take the points $$a = frac {(b + c)} {2}, b & # 39; = frac {(a + c)} {2}$$ and $$c = frac {(a + b)} {2}$$.

Now, they assume that $$d ( bigtriangleup_ {i}) = frac {d ( bigtriangleup_ {0})} {2}$$ and $$p ( bigtriangleup_ {i}) = frac {p ( bigtriangleup_ {0})} {2}$$ or $$i = 1,2,3.4$$ and $$p, d$$ represents the perimeter and the diameter of each triangle.

I was able to prove everything except that the diameter of the central sub-triangle $$bigtriangleup (a, b, c, )$$ is half the diameter of the big triangle. How could I prove it? Following the same procedure as for the three sub-triangles that share vertices with the big triangle, I can not solve anything.

Could you give me a rigorous proof of this property? Thank you so much.