1. **Problem Statement:** We have triangle $\triangle TUV$ with circumcenter $D$, where $D$ is the intersection of the perpendicular bisectors $AD$, $BD$, and $CD$. Given lengths: $$CD = 54, \quad BV = 76, \quad UD = 90.$$ We need to find the lengths of segments $$UV, VD, \text{ and } TC.$$ 2. **Key Concept:** The circumcenter $D$ is equidistant from all vertices of the triangle. This means: $$DT = DU = DV = R,$$ where $R$ is the circumradius. 3. **Using the given data:** - Since $D$ lies on the perpendicular bisector of $UV$, $D$ is equidistant from $U$ and $V$, so: $$DU = DV = 90.$$ - Given $BV = 76$ and $CD = 54$, these are segments on the sides or their bisectors, but we need to relate them to find $UV$ and $TC$. 4. **Finding $UV$:** Point $B$ is the midpoint of side $UV$ because $BD$ is a perpendicular bisector. Therefore: $$UV = 2 \times BV = 2 \times 76 = 152.$$ 5. **Finding $VD$:** From step 3, since $D$ is circumcenter: $$VD = DU = 90.$$ 6. **Finding $TC$:** Point $C$ is the midpoint of side $TV$ because $CD$ is a perpendicular bisector. Given: $$CD = 54,$$ which is the distance from $C$ to $D$ along the bisector. Since $D$ is the circumcenter, $DT = DV = 90$. Using the right triangle $TCD$ (right angle at $C$), by the Pythagorean theorem: $$TC = \sqrt{DT^2 - CD^2} = \sqrt{90^2 - 54^2} = \sqrt{8100 - 2916} = \sqrt{5184} = 72.$$ **Final answers:** $$UV = 152,$$ $$VD = 90,$$ $$TC = 72.$$