Two Circles In a Square II
The applet illustrates the following problem [Tao, pp. 66-68]:
Let ABCD be a square, and let k be the circle with center B passing through A, and let l be the semicircle inside the square with diameter AB. Let E be a point on l and let the extension of BE meet the circle k at F. Prove that
References
 |Activities| |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2012 Alexander Bogomolny
Introduce T - the intersection of AF and l. Observe that AD is tangent to both the circles at A. From here, incircle k, the angle DAF between the tangent AD and chord AF is half the central angle ABF:
2∠DAF = ∠ABF.
In circle l, for a similar reason, the angle DAT
∠DAT = ∠ABT.
It follows that ∠ABT is half ∠ABF
∠TBE = ∠ABT.
On the other hand,
∠TBE = ∠TAE,
as inscribed angles subtended by the same arc, which implies the required identity:
∠DAF = ∠EAF.
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