1, Euler formula in the fraction: a r/(a-b) (a-c)+b r/(b-c) (b-a)+c r/(c-a) (c-b).
2. Euler formula in complex variable function theory: e ix = cosx+isinx, e is the base of natural logarithm, and I is the imaginary unit.
3. Euler formula in triangle: let r be the radius of the circumscribed circle of the triangle, r be the radius of the inscribed circle, and d be the distance from the outer center to the inner center, then: D 2 = R 2-2rr.
The three forms can be understood as Euler formula, which has different meanings in different disciplines.
Prove Euler formula by mathematical induction;
1. When R= 2, from the description of 1, these two regions can be imagined as two hemispheres with the equator as the boundary, and there are two "vertices" on the equator that divide the equator into two "boundaries", that is, R= 2, V= 2, and E = 2；; So R+ V- E= 2, euler theorem is established.
Second, let R= m(m≥2) hold true for euler theorem. It is proved that euler theorem holds for R= m+ 1.
It can be seen from Note 2 that if we choose any region X on the map with R= m+ 1, then X must have such an adjacent region Y, so that after removing the unique boundary between X and Y, there are only m regions on the map.
After removing the boundaries of X and Y, if the vertices at both ends of the original boundary are still vertices of three or more boundaries, the vertices will remain, while the number of other boundaries will remain unchanged. If the vertex at one or both ends of the original boundary is now the vertex of two boundaries, the vertex is deleted, and the two boundaries on both sides of the vertex become one boundary.
Therefore, when removing the unique boundary between x and y, there are only three situations:
1, reducing an area and a boundary.
2. Reduce an area, a vertex and two boundaries.
3. Reduce one area, two vertices and three boundaries.
That is, when the boundary between x and y is removed, there must be "reduced number of regions+reduced number of vertices = reduced number of boundaries" anyway. We reverse the above process (that is, draw the boundary between x and y as it is), and it becomes a graph of R= m+ 1 again. In this process, it must be "increased number of regions+increased number of vertices = increased number of boundaries".
Therefore, if euler theorem holds when R= m (m≥2), euler theorem holds when R= m+ 1.
From one and two, we can see that for any positive integer R≥2, euler theorem holds.
Refer to the above? Baidu Encyclopedia-Euler Formula