Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.4 AIME, harder Prove it

Example 1 As shown in Figure 2.3.1, from a point $A$ outside circle $O$ , a tangent is drawn, touching the circle at $B$ . Through the midpoint $M$ of $AB$ , a secant is drawn intersecting the circle at $C$ and $D$ . Lines $AC$ and $AD$ intersect the circle again at $E$ and $F$ . Prove: $AB \parallel EF$ .

Solution

From $M A^{2}=M B^{2}=M C \cdot M D$ , we can prove $\triangle A M C \sim \triangle D M A$ , from which we have $\angle C A M=\angle C D A$ . Also, $\angle C D A=\angle C E F$ , thus $\angle C A M=\angle C E F$ (alternate interior angles), $A B / / E F$ is proven.

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