What are my dimensions? Task: Given the area of a triangle, find a Heronian triangle with that area. Any Heronian triangle with the specified area is allowed.

A Heronian triangle is a triangle with integer sides and integer area. By Heron's formula, a triangle with sides lengths a,b,c has area 

sqrt(s*(s-a)*(s-b)*(s-c))


where s=(a+b+c)/2 is half the perimeter of the triangle. This can also be written as

sqrt((a+b+c)*(-a+b+c)*(a-b+c)*(a+b-c)) / 4


If no such triangle exists, output with a consistent falsey value.

Input: A single, positive integer representing the area of the triangle.

Output: Any three side lengths for such a triangle OR a falsely value.

Examples:

Input -&gt; Output
6 -&gt; 3 4 5
24 -&gt; 4 15 13
114 -&gt; 37 20 19
7 -&gt; error


Standard loopholes apply

This is code golf, shortest answer in bytes wins.
Jelly, 17 16 bytes

-1 byte thanks to Erik the outgolfer (make use of the quick, ¥)

SHð;_P
ṗ3Ç⁼¥Ðf²Ḣ


Brute force application of Heron's formula.

Try it online! (reaches the 60s time out for the 114 tests case. Takes 3m 30s locally - it does check 1143 = 1,481,544 triples)

How?

A true golf solution - given an area a it finds all tuples of three integers between 1 and a (even with repeated triangles and ones of no area), gets their area and filters for those with the desired area (it doesn't even stop as soon as one is found, it ploughs through them all and pops the first result afterwards). Yields 0 if none exists.

SHð;_P - Link 1, get the square of the area of a triangle: list of sides
S      - sum the sides (get the perimeter)
 H     - halve
  ð    - dyadic chain separation (call that p)
    _  - subtraction (vectorises) =    [p-side1,  p-side2,  p-side3]
   ;   - concatenate              = [p, p-side1,  p-side2,  p-side3]
     P - product                  =  p*(p-side1)*(p-side2)*(p-side3)
                                  = the square of Heron's formula = area squared

ṗ3Ç⁼¥Ðf²Ḣ - Main link: number a (area)
ṗ3        - third Cartesian power (all triples of [1,area] : [[1,1,1],[1,1,2],[1,2,1],[1,2,2],[2,1,1],[2,1,2],[2,2,1],[2,2,2], ... ,[a,a,a]]
       ²  - square a
     Ðf   - filter keep if:
    ¥     -   last two links as a dyad:
  Ç       -     call last link (1) as a monad f(list of sides)
   ⁼      -     left (that result) equals right (square of a)?
        Ḣ - head - get the first one (an empty list yields 0, perfect for the falsey case)