Ptolemy's Riddle II

Geometry Level 4

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 above.

Ptolemy: Dost thou see that all the red lines have the lengths in whole integers?
Pupil: Indeed, master! Such an extra-ordinary point!
Ptolemy: Now if the equilateral triangle has side length of 13, what is the sum of the three red lengths combined?

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