Can You Solve Euler's Triangle Formula with Just the Distance (d)?

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Euler's triangle formula relates the distance (d) between the incenter and circumcenter of a triangle to the circumradius (R) and inradius (r) through the equation d² = R(R - 2r). It is established that knowing only the distance (d) is insufficient to uniquely determine both R and r, as multiple triangles can share the same distance. An example provided is the equilateral triangle, which demonstrates that one distance can correspond to various combinations of R and r. The consensus is that without additional information, it is impossible to compute both radii from just the distance. Thus, solving for R and r solely based on the distance (d) is not feasible.
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Hi all

I have a question concerning Euler's triangle formula, where the distance (d) between the incenter and circumcenter of a triangle is given as d2 = R(R-2r), with R being the circumradius and r the inradius. I suck at algebra and I need to know how to solve this if I only know the distance (d) and nothing else. How do I compute R and r from just the distance?

Many thanks
Nigel
 
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You don't. You're not going to get two values out of one. (Think of an equilateral triangle of arbitrary size for instance.)
 
Martin Rattigan said:
You don't. You're not going to get two values out of one. (Think of an equilateral triangle of arbitrary size for instance.)

Hi Martin

Thanks for the reply, I suspected that would be the case but wanted to make sure!

Cheers
Nigel
 

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