Omni-MATH / Points A, V₁, V₂, B, U₂, U₁ lie fixed on a circle Γ, in that order, and such that BU₂>AU₁>BV₂>AV₁. Let X be a variable point on the arc V₁V₂ of Γ not containing A or B. Line XA mee…

Points A, V₁, V₂, B, U₂, U₁ lie fixed on a circle Γ, in that order, and such that BU₂>AU₁>BV₂>AV₁.

Let X be a variable point on the arc V₁V₂ of Γ not containing A or B.  Line XA meets line U₁V₁ at C, while line XB meets line U₂V₂ at D.  Let O and ρ denote the circumcenter and circumradius of △XCD, respectively.

Prove there exists a fixed point K and a real number c, independent of X, for which OK²−ρ²=c always holds regardless of the choice of X.

Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.

Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$.  Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$.  Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.

Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.