I expect that the best approach is to have two circles tangentially intersecting each other, and their centers representing 'Pine' and 'Oak' trees respectively; while the intersection of two circles being the location of 'gallows'. Also, the each circles' two radius are perpendicular to each other. The other end on the perpendicular radii represents 'Spike1', 'Spike2' respectively.
I request some help in finding the intersection point of two circles that represents the gallows, as the main issue is this configuration is not workable , i.e. the intersection of two circles is meaningless once the relative angle among the pine and oak trees (the two radii) is changed.
The only way is for the two circles to be intersecting still, but be able to to move while intersecting at more than one point possibly, as in the below diagram.
Anyway, the approach should ensure the independence of treasure from the position of the gallows, as shown in the first link. also, may be for the above approach using circles, polar coordinates' equations may work to represent using coordinate geometry in desmos.
Would prefer the circle based approach, as above.
An alternate approach as shown in the first link also, is to place the trees on the x-axis, with gallows changing location along the locus of some parametric equation as a function of distance from the trees.
In the first approach, the movement of gallows is not possible, and only a particular configuration can be shown like in the first / second diagrams, while in the second approach (as shown in the third diagram, below) the gallows moves as a function on locus.