Math CentralQuandaries & Queries


Question from Murtaza:

Two circles touch externally at T. A chord of the first circle XY is produced and touches the other at Z. The chord ZT of the second circle, when produced, cuts the first circle at W. Prove that angle XTW = angle YTZ.

We have two responses for you


Draw an accurate diagram. There are certain important points (the centers of the circles) that have not been mentioned. How do they relate to T? Add some relevant lines.

Now, you are trying to show angles are congruent. There are various ways of doing this (congruent triangles, Star Trek lemma, similar triangles, opposite angles, angle sum theorems, parallel lines...) Which do you think might apply here?

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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