Scale the triangle so the longest side has measure 1.

Take a unit length line segment and construct a semicircle about each end point of radius (sqrt(5) + 1) / 2.  Any third vertex such that it is within 1 unit of both endpoints of the unit segment will be within (sqrt(5) + 1) / 2 of each endpoint and therefore its ratio will be below that value, but there will be a small area near the midpoint of the unit segment where the distance from each endpoint will be less than (sqrt(5) - 1) / 2, as that number is slightly over .5.  What's necessary is to prove that the ratio of the distances to the two endpoints will be in the required range, relative to each other.

The longer of the two variable sides in that area will be .5 <= d1 <= (sqrt(5) - 1) / 2 while the shorter of the two variable sides will be 1 - (sqrt(5) - 1) / 2 <= d2 <= .5. The largest ratio of longer to shorter will be ((sqrt(5) - 1) / 2) / (1 - (sqrt(5) - 1) / 2) and the smallest ratio is .5 / .5 = 1. That largest amounts to (sqrt(5) + 1) / 2, but only happens when the variable point is on the perpendicular bisector of the unit segment so that the ratio of either variable segment to the unit segment is (sqrt(5) - 1) / 2, assuring that the strict equality can be applied to just that value.