Isogonal Lemma

What Might This Be About?

Problem

Points $D,$ $E,$ $F,$ $G,$ $Q$ are on the circumcircle $(ABC)$ such that $\angle BAF=\angle FAC,$ $\angle BAD=\angle QAC,$ $\angle ABG=\angle GBC,$ and $AD\parallel BE.$

Isogonal Lemma - problem

Then $\angle EBG=\angle GBQ.$

Proof

The proof is by simple angle (or, perhaps, more appropriately, arc) chasing.

Isogonal Lemma - problem

Since $AD\parallel BE,$ $BD=AE.$ Now, in terms of arcs, it is given that $BD=CQ$ and $AG=CG.$ From here, $EG=GQ$ and, therefore $\angle EBG=\angle GBQ.$

The lemma could be reformulated as follows:

Let $Q\in (ABC)$ and $AD$ is the isogonal conjugate of $AQ$ in $\angle BAC.$ Let $BE\parallel AD.$ Then $BE$ is the isogonal conjugate of $BQ$ in $\angle ABC.$

Replacing $B$ with $C,$ drawing a parallel to $AD$ through $C$ and its isogonal conjugate with respect to $\angle ACB,$ we see that the latter too passes through $Q.$ For this reason, $Q$ could be said to be the isogonal conjugate of the point at infinity corresponding to the direction of $AD.$

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