During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks.
Given a circle $(O, R)$ and $M$ is a point outside the circle. From $M$, we draw two tangents $MA$ and $MB$ and a secant $MEF$ with the circle $(O)$ ($E$ is between $M$ and $F$). Draw a diameter through $B$ of $(O)$ with the other endpoint is $D$. $BD$ intersects with $EF$ at $G$, and $DA$ intersects with $EF$ at $I$. Prove that $(EFGI) = -1$
Here is the picture:
So I can solve this problem, but I feel like it's too complicated. My solution is: I denote the intersection of $OM$ with $DE$ and $DF$ are $P$ and $Q$, and I try to prove that $OP = OQ$ by proving that $DAPQ$ is an isosceles trapezoid (please let me know if anyone needs my full answer). I feel like my solution is too manual. But I haven't found any other solution. The difficulty here lies in the definition of $G$, which I can't find its relation with other point. Please help me with it. I really appreciate.
One more thing: It seems interesting for me that when the secant moves but always passes $M$, we always have the same property. When we fix $M$, we have 2 lines fix: $DA$ and $DB$. So I wonder if there is any notion like "2 lines harmonic with a circle" like the notion of "harmonic pencils" created by four lines.