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 1 of 5: Identify the two equilateral triangles

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\triangle ABC\text{ and }\triangle ACD\text{ are equilateral}

Detailed analysis

Since $AB=BC$ and $\angle ABC=60^\circ$ , triangle $ABC$ is equilateral, so $AC=AB$ . Because all sides of the quadrilateral are equal, $AD=CD=AC$ , and triangle $ACD$ is also equilateral.