I'm a bit stuck on the question below, and I wondered if anyone out here might be able to help:
Construct a circle with a centre in O(0,0) and a radius of 5. Two tangents of the circle intersect in point B(13,0). The tangents touch the circle in points C and D. There is a point E on the circle's perimeter, and the tangent in point E intersects the other tangents in points F and G.
i) Examine the perimeter of the triangle GBF when E is moved along the arc of the circle between C and D. (I did this, and the perimeter is constant (=24). But why?
ii) Prove a relationship between the length of EF and the perimeter of GBF. It may be useful to determine values for the length of EF and the perimeter of GBF when E is on the x axis. (I've found: EF=10/3, perimeter(GBF)=24)
Image:
You want to know the perimeter of the triangle, which is $p = BG + FG + BF$. If you partition $FG$ further into $EF + EG$, we're getting closer.
Here is the kicker: Since the segments $GE$ and $GD$ are both tangents to the circle from the same point $G$, they are equally long. For the same reason, we have $FE = FC$. If we substitute that in our sum, we get $$ p = BG + FG + BF = BG + EF + EG + BF = BG + GD + FC + BF $$ which you can see is completely independent on where $E$ is placed, it's just the length of the two segments $BC$ and $BD$.