Fang-jh wrote: Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value. Let $ l_1$ , $ l_2$ be perp. bisectors of $ AD,BC$. let $ l_3$ , $ l_4$ be perp bisector of AE and EB. then $ O_1O_2$= distance between intersection of$ (l_1,l_3)$ and $ (l_2,l_4)$. since $ l_1,l_2$are fdixed and perp. dist.of ($ l_3,l_4$)=$ \frac{AB}{2}$,our result follows.

ith_power wrote: Fang-jh wrote: Given a trapezoid $ ABCD$ with $ AD\parallel BC, E$ is a moving point on the side $ AB,$ let $ O_{1},O_{2}$ be the circumcenters of triangles $ AED,BEC$, respectively. Prove that the length of $ O_{1}O_{2}$ is a constant value. ith_power wrote: Let , be perp. bisectors of . let , be perp bisector of AE and EB. then = distance between intersection of and . since are fdixed and perp. dist.of ()=,our result follows. Let $ l_1$ , $ l_2$ be perp. bisectors of $ AD,BC$. let $ l_3$ , $ l_4$ be perp bisector of AE and EB. then $ O_1O_2$= distance between intersection of$ (l_1,l_3)$ and $ (l_2,l_4)$. since $ l_1,l_2$are fdixed and perp. dist.of ($ l_3,l_4$)=$ \frac{AB}{2}$,our result follows. I found the distance between the circumcenters is $(AB/2).cosec(ADC)$