# Circle Geometry Question Involving Two Tangents and a Set of Parallel Lines

Tangents $OA$ and $OB$ are drawn to a circle from an external point $O$. Through the point $A$, a chord $AC$ is drawn parallel to the tangent $OB$ and $OC$ passes through the circle at $E$.

I am required to show that A$F$ bisects $OB$.

My approach thus far has been to observe that triangles $AFO$ and $OFE$ are similar, and also $AEB$ is similar to $BEO$.

• I am required to show hat AF bisects OB. So yes, equivalently F would be the midpoint of BO. – Hugh Entwistle Jul 22 '18 at 9:48
• Math.SE demands you to show at least some work on your question. What have you tried so far? – Cargobob Jul 22 '18 at 9:50
• I detailed above, that I have been able to prove that Triangles AFO and OFE are similar. And that triangles ABE and EBO are similar. Working out was omitted, but I can give you the working if you so desire. – Hugh Entwistle Jul 22 '18 at 9:52

Since $\angle EAO = \angle EOF$ we see that the line $FO$ is tangent to circumcircle $(AEO)$, so by PoP with respect to $F$ and $(AEO)$ we have $$FO^2 = FE\cdot FA$$
and by PoP on a given circle we have $$FB^2 = FE\cdot FA$$
so $FB = FO$.