AoPS - Problem

Given

$\triangle ABC\ :\ A(0,a),B(-b,0),C(c,0)$ . Incenter

$U(\frac{cu-bv}{b+c+u+v},\frac{a(b+c)}{b+c+u+v})$ with

$u=\sqrt{a^{2}+b^{2}}$ and

$v=\sqrt{a^{2}+c^{2}}$ . Circumcenter of

$\triangle BCU\ :\ O_{1}(\frac{c-b}{2},\frac{[a^{2}-(b+u)(c+v)](b+c)}{2a(b+c+u+v)}\ )$ . Circumcenter of

$\triangle ACU\ :\ O_{2}(\frac{a^{2}+c^{2}-bv+c(2u+v)+uv}{2(b+c+u+v)},\frac{a^{2}(b+2c+u+v)-c(b-u)(c+v)}{2a(b+c+u+v)}\ )$ . Circumcenter of

$\triangle ABU\ :\ O_{3}(-\frac{a^{2}+b^{2}+b(u+2v)-cu+uv}{2(b+c+u+v)},\frac{a^{2}(2b+c+u+v)+b(b+u)(v-c)}{2a(b+c+u+v)}\ )$ . Distance

$OO_{1}=OO_{2}=OO_{3}=\frac{uv}{2a}$ .