definition
In a polygon, the point with the smallest sum of distances to each vertex is called the fermat point of the polygon.
In a plane triangle:
(1). Three triangles with internal angles less than 120, with AB, BC and CA as sides, make regular triangles ABC 1, ACB 1, BCA 1 on the outside of the triangle, and then connect AA 1.
(2) If the internal angle of the triangle is greater than or equal to 120 degrees, then the vertex of this obtuse angle is the demand.
(3) When △ABC is an equilateral triangle, the outer center coincides with fermat point.
(1) In an equilateral triangle, BP=PC=PA, and BP, PC and PA are the heights of the three sides of the triangle and the bisector of the triangle respectively. Is the center of inscribed circle and circumscribed circle. △BPC?△CPA?△PBA .
(2) When BC=BA but CA≠AB, BP is the bisector of the height and the median line on the triangle CA and the angle on the triangle.
certificate
(1) The opposite opening angle of fermat point is 120 degrees.
△CC 1B and △AA 1B, BC = ba 1, Ba = bc 1, ∠ CBC 1 = ∠ b+60 degrees = ∠ ABA/kloc.
△CC 1B and △AA 1B are congruent triangles, and ∠PCB=∠PA 1B is obtained.
In the same way, ∠CBP=∠CA 1P can be obtained.
From ∠PA 1B+∠CA 1P=60 degrees, ∠PCB+∠CBP=60 degrees, so ∠CPB= 120 degrees.
Similarly, ∠APB= 120 degrees, ∠APC= 120 degrees.
(2)PA+PB+PC=AA 1
Rotate △BPC 60 degrees around point B to coincide with △BDA 1 and connect PD, then △PDB is an equilateral triangle, so ∠BPD=60 degrees.
And ∠BPA= 120 degrees, so a, p and d are on the same straight line.
And ∠CPB=∠A 1DB= 120 degrees, ∠PDB=60 degrees, ∠PDA 1= 180 degrees, so a, p, d, a.
(3)PA+PB+PC is the shortest.
Take any point M (not coincident with point P) in △ABC, connect AM, BM and CM, rotate △BMC 60 degrees around point B to coincide with △BGA 1, connect AM, GM, A 1G (same as above), and then AA 1
Plane quadrilateral fermat point
Fermat point's proof in a quadrilateral is easier to learn than fermat point's proof in a triangle.
(1) In the convex quadrilateral ABCD, fermat point is the intersection point p of two diagonal lines AC and BD.
(2) In the concave quadrilateral ABCD, fermat point is the concave vertex D(P).