It is most convenient to solve this problem with coordinate system.
As shown in the figure, A, B and C distinguish beech from two oak trees. I take the midpoint of two oak trees, that is, the midpoint of bc, as the coordinate origin, and A is an arbitrary point. According to the meaning of the topic, bd is perpendicular to ab, ac is perpendicular to ce, ab=bd, ac=ce.
De is the point 1/2, and let f be their midpoint, which is the point confirmed in the initial positioning mode of the treasure; On the Y axis, according to the topic, extend the distance of gb or cg (the same anyway) down to H, which is the second positioning method.
If it can be proved that the coordinate of H is F, then the problem will be solved.
Suppose the length of ab is m, the length of ac is m, the length of bg is p, and the angle abc is α and the angle acb is β. The vertical legs passing through with axis A, D and E are h 1, h2 and h3 respectively.
So according to the original method of pirates, according to congruent triangles or something, the angle bdh 1 is α, and the angle ceh3 is β, right?
h 1b = BD * sinα= ba * sinα= m * sinα
The length of h 1g is p-m*sinα.
h 1d=m*cosα
So we can deduce that the D coordinate is (-p+m*sinα, -m*cosα) and the E coordinate is (p-n*sinβ, -n*cosβ).
According to the midpoint coordinate formula of two points, we can know that the coordinate of point F is ((a * sinα-b * sinβ)/2, -a * cosα+b * cosβ)/2).
According to the second piracy method, the H coordinate is (0, -p).
Next, just prove that (a * sinα-b * sinβ)/2 = 0; ，-(a * cos α+b * cos β)/2 =-p。
In triangle abc, we know from sine theorem that a * sinα = b * sinβ (a/Sina = b/sinb = c/sinc = constant), so obviously (a*sinα-b*sinβ)/2=0 holds.
And BC = 2bg = 2p = bh2+h2c = ab * cosα+AC * cosβ = a * cosα+b * cosβ.
So we know that p=(a*cosα+b*cosβ)/2, and the ordinate is also proved.
To sum up, the coordinates of point H and point F coincide, that is, point H and point f***, that is, the two search methods of pirates can indeed find the same point.
The biggest advantage of using coordinates is that you can avoid all kinds of boring reasoning in junior high school, such as auxiliary lines and congruences. Putting them in the coordinate system is calculating the coordinates. Advanced methods always simplify the process. Thanks to Descartes, this rectangular coordinate system was first created by Descartes, also called Cartesian coordinate system ~ ~