Euler's theorem in geometry

In the geometry of the set of Euler supplies, named after Leonhard Euler, a formula for the distance between the center points of radius and incircle of a triangle.

It denotes the circumradius and the Inkreisradius.

From the theorem Euler's inequality immediately follows:

Evidence

There are the circumcenter and incenter of the triangle. The line intersects the bisector according to the South Pole Set the radius to a point which lies on the corresponding bisectors. The second point of intersection of the perpendicular bisectors () with the radius is. Denoting the base of the felled by from solder to with, then apply.

Because of compliance in two angles, the triangles are similar to each other and. Therefore, applies and on. Thus we have shown:

If you connect with, so you can use the exterior angle theorem, as large as the two non- adjacent interior angles after an exterior angle () of a triangle ():

Using the peripheral angle theorem Moreover, it follows

From which it follows. Triangle is isosceles so; it applies. From what has been proved already obtained

Now and are the intersections of the line with the circle. Application of the law resulted in tendons

The track lengths on the left hand side can be expressed by the radius and the radius distance:

Through a brief transformation, one obtains the assertion:

Related statement

If the radius of the side associated excircle, then for the distance between the center of this excircle and the circumcenter:

The same applies for the other two excircles.

Weblink

• Eric W. Weisstein: Euler Triangle Formula. In: MathWorld (English).
• Euler's theorem - a visualization with GeoGebra.

• Triangle geometry
• Set ( mathematics)