Let AE, AD and DE be X, Y and Z respectively, and find the minimum value of Z. ..

S(ADE)= S(ABC)/2 = 5 * 12/4 = 15

s(ade)=( 1/2)xy(Sina)=( 1/2)xy *(5/ 13)= 15。

So xy=78, then x2+y2 >；; =2xy= 156

So according to the cosine theorem:

z^2=x^2+y^2-2xy(cosa)=x^2+y^2-2*78*( 12/ 13)=x^2+y^2- 144>； = 156- 144= 12

So z & gt=2 times the root number 3, and x=y= the root number 78 when the equal sign holds.

That is, the minimum length of DE is twice that of root number 3.