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Right triangle ABC, points D and E are in the side of BC. AD=AC and BE=AB. How much is angle DAE?

 
In , and CD is of angle C. Prove CA+DA=BC.

Rectangle ABCD, P is in the side of BC and Q is in the side of CD. PA intersects PQ in 45 degree. Prove BP+DQ=QP.

 
Circles O1 and O2, their chord AB=8, calculate the distance between their centers (O1O2= ?).
In circle O, AB and CD are diameters, chord EF is vertical for OC and goes halves in OC. Prove .
A triangle ABC, side AB=a, the height on side AB is h. Rectangle EFGH is inscribed, see image. How height is HE, the area of rectangle is maximum?  (Plot triangle ABC in setting AB=3 and its height = 3)