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Ken's Puzzle of the Week

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A Set of Triangles

In
rectangle ABCD, point E is on AB and point F is on AD. Angle
CEF is a right angle. Draw triangle CEF.
The rectangle is divided into four triangles (AEF, BEC, DCF,
CEF). All
three sides of each triangle are integers, and CF is the longest line
in the
diagram. What is the smallest possible length of CF?

Extension: What solutions
exist for CF < 100? In particular what solutions exist which
aren't multiples of smaller solutions?

Source: Original.

Solution

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to Ken