Force problem using springs

[GoGoGadget]

Homework Statement

2. You have been asked to evaluate a new device
designed to hold arteries open. The design has three
identical very small spheres connected in an equilateral triangle by three identical small springs. When inserted into an artery, the springs are compressed. To determine their safety, you have been asked to calculate the force exerted on the artery wall by one of the spheres as
a function of the spring constant,the compression distance of each spring, and the angle
between the sides of the triangle. For the first calculation you decide to ignore the blood flowing through the artery,the weight of the spheres and springs, and assume that the cross-section of the artery is a circle.

Homework Equations

F=ma

F=kx

cos θ = adj/hyp

sin θ = opp/hyp

The Attempt at a Solution

See attached document. I understand the setup for the problem in that we looking for the force of the spring on the artery. I also understand why Newton's third law of motion would be incorporated as the force of the sphere is the force of the artery and vice versa and that all objects are in equilibrium so the sum of forces in both the +x and +y directions are zero. My question, why in the diagram does it make sense to divide the angles of the triangle in half? Is it due to the opposing motion of the Fa? This is what I mostly don't understand about the problem...

 

[haruspex]

The coordinates have been chosen so that the y-axis bisects the angle. This makes sense since the symmetry implies the forces in the springs are equal, so the resultant will lie on the angle bisector.