The length of AB is fixed, that is, find the minimum value of PA+PB.

On the Axisymmetric A of X to (2,3)

The intersection of the straight line passing through (2,3) and (4,-1) and the X axis is p.

Y=-2X+7, when Y=0, X=7/2, then P (7/2,0).

2

The lengths of AB and CD are fixed, that is, find the minimum value of AC+BD.

Translate B to the left by 3 units to (1,-1), and then be symmetrical about X axis to (1, 1).

C is the intersection of the straight line passing through (1, 1) and (2, -3) and the X axis.

Y=-4X+5, when Y=0, X=4/5, then C (4/5,0).

three

The length of AB is fixed, that is, find the minimum value of AN+MN+BM.

Symmetrize a to (-2,3) about the origin.

The intersection of the straight line passing through (-2,3) and (4, 1) and the X axis is m.

Y=-2X/3+5/3, when Y=0, X=5/2, then M (5/2,0).

AN is parallel to BM, then n-(-3): 2 = 1: 4-5/2.

Get N=-5/3.