Sangaku again

A circle (AB) with diameter AB
A circle (AC) with diameter AC, from any C on the diameter AB
The isosceles triangle BCD, with D on the first circle
And now pure magic :
The center of circle touching (AB), (AC) and the side CD has an amazing property !
When (for which choice of point C) does this center lie on the perpendicular to AB from C ? Solution

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