Fermat

Fermat, Pierre de Fermat (160 1 ~ 1665), a French mathematician, is known as the "king of amateur mathematicians". His father Dominic Fermat opened a large leather goods store in the local area, and the industry was very rich, which made Fermat live in a rich and comfortable environment since he was a child. Toulouse

Fermat point definition

In a triangle, the point with the smallest sum of the distances to the three vertices is called the fermat point of the triangle. (1) If all three internal angles of the triangle ABC are less than 120, then the three distance lines just bisect the rounded corner where fermat point is located. So the fermat point of a triangle is also called the isocenter of a triangle. (2) If the internal angle of a triangle is not less than 120 degrees, the vertex of this obtuse angle is the point with the smallest distance sum.

Edit fermat point's judgment in this paragraph.

(1) For any triangle △ABC, if there is a little E in or on the triangle, if EA+EB+EC has the minimum value, then E is fermat point. Fermat point's calculation

(2) If the interior angle of a triangle is greater than or equal to 120, the vertex of this interior angle is the fermat point; If all three internal angles are less than 120, then three points with an opening angle of 120 inside the triangle are the fermat point of the triangle.

Edit this paragraph to prove

How to Prove fermat point: fermat point Proof Graph.

(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. Get ∠ PCB = ∠ pa1b ∠ CBP = ∠ ca1p ∠ pa1b+∠ ca1p = 60 degrees. ∠APC= 120 degrees (2)PA+PB+PC=AA 1 Rotate △BPC around point B by 60 degrees to coincide with △BDA 1 and connect PD, then △PDB is an equilateral triangle, so △ BPD = 67. And ∠CPB=∠A 1DB= 120 degrees, ∠PDB=60 degrees, ∠PDA 1= 180 degrees, so a, p, d, a. (3)PA+PB+PC randomly selects a point M (not coincident with point P) in the shortest △ABC, connects AM, BM and CM, rotates △BMC 60 degrees around point B to coincide with △BGA 1, connects AM, GM, A 1G (same as above), and then AA/Kloc. A 1G+GM+MA=AM+BM+CM。 So the distance from Fermat to the three vertices of A, B and C is the shortest. The proof of fermat point in fermat point quadrilateral is simpler and easier to learn than that of fermat point in triangle. (1) In the convex quadrilateral ABCD, fermat point is the intersection point p of two diagonal lines AC and BD. fermat point

(2) In the concave quadrilateral ABCD, fermat point is the concave vertex D(P). After the above deduction, we can find the fermat point in the triangle: when the internal angle of the triangle is greater than or equal to 120 degrees, fermat point is the vertex of this internal angle; If all three internal angles are within 120 degrees, then fermat point is the point that makes the connecting line between fermat point and the three vertices of the triangle form an included angle of 120 degrees.

Edit the fermat point nature of this paragraph:

fermat point

On the (1) plane, the sum of the three vertices from point P to △ABC is PA+PB+PC, and when point P is fermat point, the sum of the distances is the smallest. Among the special triangles: (2) Three triangles with internal angles less than 120, with AB, BC and CA as sides respectively, make regular triangles ABC 1, ACB 1 and BCA 1 on the outside of the triangle, and then connect AA 1 and BB6544. Then point P is the fermat point sought. (3) If the internal angle of the triangle is greater than or equal to 120 degrees, the vertex of this obtuse angle is the vertex to be found. (4) When △ABC is an equilateral triangle, the outer center coincides with fermat point! ! ! This is the information. Write it yourself.