Once upon a time, Ptolemy let his pupil draw an equilateral triangle \(ABC\) inscribed in a circle before the great mathematician depicted point \(D\) and joined the red lines with other vertices, as shown below.

**Ptolemy**: Dost thou see that all the red lines have the lengths in whole integers?

**Pupil**: Indeed, master! Such an extraordinary point!

**Ptolemy**: Now if the equilateral triangle has a side length of 13, what is the sum of the three red lengths combined?

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