Math CentralQuandaries & Queries


Question from Murtaza:
Line ATB touches a circle at T and TC is a diameter.
AC and BC cut the circle at D and E respectively.
Prove that the quadrilateral ADEB is cyclic.

We have two responses for you



  1. Note that "touches" means "is tangent to". Note that "cyclic" is equivalent to "one pair of opposite angles are supplementary".

  2. As always construct and label a diagram. There are two obvious missing segments not referred to in the problem - add them in too.

  3. As you proceed, find and label whatever angles you can.

Good Hunting!



Before attempting to prove your result you should first try to understand the Euclidean theorem that describes the relationship between, on the one hand, the angle between a chord of a circle and the tangent to the circle at one of the chord's endpoints, and on the other hand, an angle inscribed in the circle that is subtended by that chord. The best way to understand the theorem is to draw a few accurate figures showing a circle with a line AT tangent to it at T, and points C and D taken at various places about the circle. You should compare the angles ATD and TDC. Try drawing cases where angle ATD is acute, right, and obtuse; in each case, choose C on the larger of the two arcs TD and again on the smaller arc. After investigating these six cases, you should find the proof of your result quite easy.


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