When all three internal angles of the triangle are less than 120 degrees, and point P meets the requirements of 120 degrees for PA, Pb and PC respectively, PA+Pb+PC has a minimum value. At this time, it is obvious that point P should be the heart of a regular triangle, and Pa+Pb+PC = root number 3.

The following is the proof of the theorem for your reference:

Fermat point refers to the point where the sum of the distances from the three vertices of the triangle in the plane is the smallest. (1). The fermat point of three triangles with internal angles less than 120, with AB, BC and CA as sides respectively, makes the regular triangles ABC 1, ACB 1 and BCA 1 on the outside of the triangle. BB 1, CC 1, then the three lines intersect at a point p, then the point p is the fermat point sought. (2) If the internal angle of a triangle is greater than or equal to 120 degrees, then the vertex of this angle is the vertex to be found.

For any triangle △ABC, if a point P in the triangle makes the three line segments of PA+PB+PC have the minimum value, P is fermat point.

The way of doing things

* When the internal angles of triangles are all less than 120 degrees.

O make three regular triangles △ABC', △BCA' and △CAB' outward.

O connect CC', BB', AA'

* When the internal angle is not less than 120 degrees, fermat point is the vertex corresponding to this angle.