Euler Triangle Formula/Lemma 1

Lemma to Euler Triangle Formula

Let the incenter of $\triangle ABC$ be $I$.

Let the circumcenter of $\triangle ABC$ be $O$.

Let $OI$ be produced to the circumcircle at $G$ and $J$.

Let $CI$ be produced to the circumcircle at $P$.

Let $F$ be the point where the incircle of $\triangle ABC$ meets $BC$.

We are given that:

the distance between the incenter and the circumcenter is $d$

the inradius is $\rho$

the circumradius is $R$.

Then

$IP \cdot CI = \paren {R + d} \paren {R - d}$

Proof

$OI = d$

$OG = OJ = R$

Therefore:

$IJ = R + d$

$GI = R - d$

By the Intersecting Chords Theorem:

$GI \cdot IJ = IP \cdot CI$

Substituting:

$IP \cdot CI = \paren {R + d} \paren {R - d}$

$\blacksquare$