Special topic: calculation problems

Analysis: In this problem, we must first make clear where the point P is. According to Pythagorean theorem, the length of MN can be obtained through the symmetry point M' of M about AC, and the lengths of AB, BC and AC can be obtained respectively according to the properties of midline and trigonometric function, thus the circumference of △ABC can be obtained.

Solution: let m' be the symmetrical point of point m about AC and connect M'N, then the intersection with AC is the position of point p,

∵M and n are the midpoint of AB and BC, respectively,

∴MN is the center line of △ABC,

∴MN∥AC，

∴pm′/？ PN？ = KM′/？ KM？ = 1,

∴pm′=pn，

That is, when PM+PN is the minimum, p is located at the midpoint of AC,

∴MN= 1/？ 2? Alternating current

∴PM=PN=2,MN=2？ Root number 3∴AC=4 root number? 3? ,

AB=BC=2PM=2PN=4，

∴△∴△abc's circumference is: 4+4+4? 3? =8+4? Root number? 3? .

So the answer is: 8+4? 3? . ?