let radius of circle with center $H$ is $R$. Because of $M_b,M_c$ midpoint s we have $M_bM_c$ parallel to $BC$ so $M_bM_c$ is perpendicular bisector of altitude from $A$. let $P=M_bM_c\cap AH,Q=AH\cap BC$ so we have $AP=PQ$. Then we have $AP^2-PH^2=(AP-PH)(AP+PH)=(PQ-PH).AH=AH.HQ$. If we do pisagor theorem we have $AUV_1^2-V_1H^2=AP^2-PH^2=AH.HQ$ so $AUV_1^2=V_1H^2+AH.HQ$. And $AUV_1^2=R^2+AH.HQ$. let use power in circles $(ABQR),(BCRS),(CASQ)$. let $S=CH\cap AB,R=BH\cap CA$. So we have $AH.HQ=BH.HR=CH.HS$. Similarly we have $BW_1^2=BW_2^2=R^2+BH.RH$, $CU_1^2=CU_2^2=R^2+CH.SH$, $AU_2^2=R^2+AH.QH$. So we have $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$ and we getresult....