$ 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.