Euler's Geometry Puzzle
JavaScript (ES7),  80 74 66  65 bytes

(a,b,c)=&gt;(s=a+b+c,(p=a*b*c/s)*p/4*(s/=2)/(s-a)/(s-b)/(s-c)-p)**.5

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How?
This is derived from:

The semiperimeter \$s\$ of the triangle:
$$s=\frac{a+b+c}{2}$$

The circumradius \$R\$ of the triangle:
$$R=\frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}$$

The product of the inradius \$r\$ and the circumradius:
$$rR=\frac{abc}{2(a+b+c)}=\frac{abc}{4s}$$

Euler's theorem:
$$d=\sqrt{R(R-2r)}=\sqrt{R^2-2rR}=\sqrt{R^2-\frac{abc}{2s}}$$