Problem 1

Let

ABCD

be a quadrilateral with all sides equal and

\angle ABC=60^\circ

. Let

\ell

be a line through

D

that does not meet the quadrilateral except at

D

. Let

E,F

be the intersections of

\ell

with

AB,BC

, respectively, and let

M=CE\cap AF

. Prove that

CA^2=CM\cdot CE

Step 5 of 5: Apply the tangent–secant power theorem

Previous step Next step

CA\text{ is tangent to }(AME)\Longrightarrow CA^2=CM\cdot CE

Detailed analysis

By the tangent–chord theorem, the equality $\angle CAM=\angle AEC$ proves that $CA$ is tangent to the circumcircle of $AME$ at $A$ . The line $CE$ is a secant through $C$ , meeting that circle at $M$ and $E$ . Therefore the power of $C$ gives $CA^2=CM\cdot CE$ .