Last time we talked about the mathematical model of "Hu No Return", today we will talk about the problem of "Fermat point".
First, let's take a look at what a "Fermat point" is:
Fermat, a master mathematician of the 17th century, is arguably the greatest French mathematician of the 17th century, who solved the problem of confusing the world's wise men for 358 years, and Fermat once proposed an interesting problem about triangles: find a little P in △ ABC, so that the value of PA + PB + PC is the minimum, and people call this point "Fermat point".
Mathematical model of the Fermat point
(1) As shown in Figure (1), there is a little P inside the △ ABC, connect pa, PB, PC, and find the minimum value of PA+PB+PC.
Solution idea: As shown in the figure, the △APC is rotated 60 ° around the point C, and the △EDC is obtained, and PD and BE are connected, then PA+PB+PC=ED+PB+PD BE, so when B, P, D, E are four-point collinear, PA+PB+PC gets the minimum value, and the minimum value of PA+PB+PC is BE.
(2) As shown in Figure (2), in Rt△ABC, ∠ACB = 90 °, ∠ABC = 30 °, point O is a point within △ABC, and the minimum value of AO + BO + CO is.
【Example 1】As shown in the figure, in isosceles Rt△ABC, ∠ABC = 90 °, AB = 4, point P is a point inside △ ABC, then what is the minimum value of AP + BP + CP?
【Example 2】In order to welcome the 17th Shaanxi Provincial Games held in Yulin from August 6 to 11, 2022, when decorating a sports venue, a decoration company needs to design a quadrilateral plate ABCD as shown in the figure, requiring AD//BC, AB+BC = 6 meters, ∠ABC = 60 °, point P is a point within the quadrilateral ABCD, whether there is a point P to meet ∠APC = ∠ BAD, and the sum of the distances A, B and C of the three vertices of the quadrilateral plate ABCD (that is, PA +PB+). PC) minimum? If present, the minimum value of PA+PB+PC is found; If it does not exist, please explain why.
The problem of "Fermat point" is actually relatively simple, as long as you understand his model and analyze his question type, it is often not difficult to get full marks. Today we share the problem of "Fermat point" here, and next time we share the problem of "general drinking horses".