The second question:

1, with BP as one side, making an equilateral triangle BPD；; Upward; Then take BD as the edge and make triangle BDA 1 to the left, which are all equal to triangle BPA；; Available: BA 1=AB=BC=4, pa+Pb+PC = PD+da1+PC;

2. because when A 1, d, p and c are four lines, the line segment A 1C is the shortest, and A 1C=PA+PB+PC, so the length of A 1C is the demand.

The calculation is very complicated. Let me give you an idea:

According to the above proof conclusion-the triangle BCA 1 is an isosceles triangle, and the top angle 150 degrees-the bottom angle 15 degrees-the angle PAC= the angle PCA=30 degrees. If the intersection point P is perpendicular to BC and the vertical foot F, then PF = BF and PF=BF=x, then FC=4-x, X and BP can be obtained by Pythagorean theorem, and a conclusion can be drawn.

I wish you good grades!