I'm searching for a simple expression for the distance CD in relation to BE (=2BD) in the picture above.

1) D is midpoint on BE. BE and hence BD = DE are known.

2) Angles BAC = CAE = BAE/2, that is,

**C represents the "angular midpoint" on BE** as "seen from A". Those angles are known.

3) F is the center of a circle. Points A, B and E lie on the perimeter of that circle so AF = BF is its radius.

4) Angles BFD = BAE are known. Hence triangle BDF is completely known (because of right angle and BD are also known).

Number 4 above is actually necessary because the circle describes all positions of point A which yield constant angle BAE. (See

PlanetMath: circumferential angle is half the corresponding central angle)

**I would be very interested in knowing if there exists a similar simple relationship for all points A such that instead CD remains constant!**
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