Euler Triangle Formula/Lemma 1
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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:
- $IP \cdot CI = \paren {R + d} \paren {R - d}$
$\blacksquare$