Home > Geometry, Problem Solving > Proving Euler’s Relation using inversions

## Proving Euler’s Relation using inversions

Determine the locus of the poles of all the inversions that transform two secant lines into circles of the same radius. Deduce what are the poles of inversions that transform the three sides of a triangle into circles of the same radius. Prove that the inversions about these poles transform the cicrumcircle of the triangle into a circle with the same radius as the other three circles. ( This is simple, just use this )

Now consider $ABC$ a triangle, $O$ its circumcenter, $R$ the radius of its circumcircle, $I$ its incenter, $r$ the radius of its incircle. Prove Euler’s relation $OI^2=2 r R$ in two ways, using inversions. One time, take an inversion of pole $I$ preserving the incircle, and the second time, take an inversion of pole $I$ which preserves the circumcircle.

The figure below can give you a hint of what is going on.

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Categories: Geometry, Problem Solving
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