Counting the number of intersection points in  the diagram is a little bit trickier.

计算图中的交点数手一些。

Set the intersection points equal, the equations equal to each other and solve.

学生：使交点相等，方程式相等并求解。

Then compare these two and find the intersection point, which is the determined opportunity point.

然后比较这两者，找出它们的交集点，这个交集点就是确定的机会点。

No other choice of two intersection points is guaranteed to get all the robo-ants within five minutes.

没有其他选择的两个交点可以保证在五分钟内得到所有的机器蚂蚁。

Every intersection point corresponds  with some quadruplet of points.

个交点对应于某四个点的组合。

The one that I landed on just because I feel like both of these teams are just at interesting intersection points.

最终选择这个，是因为觉得这两支球队正处在有趣的交汇点上。

But the key is to treat this as a new graph  where those intersection points are them vertices.

但关键是将这视为一个新的图，其中那些交点就是顶点。

So, in effect,   each intersection point adds two more edges.

因此，个交点实际上增加了两条边。

I claim that that line also passes through the intersection point between the external tangents of the circles that we care about.

那条线也经过们关心的圆的外切线的交点。

Now that we have a set of intersection points on separate planes, we find whether the point is inside each corresponding triangle.

现在们有一组位于不同平面上的交点，们需判断这些点是否在各自对应的三角形内部。

One idea might be that it should be the number of pairs  of chords since every intersection point comes from two different chords.

一个想法可能是它应该是弦的对数，因为个交点都来自两条不同的弦。

So the total number of vertices here  would not be N but N plus N choose four total intersection points.

所以这里的顶点总数不是N，而是N加上n选4的总交点数。

For example, look at this simple diagram where  we have three lines and two intersection points.

例如，看这个简单的示意图，们有三条线和两个交点。

The claim, the theorem, the thing you need to prove is that those three intersection points of the external tangents always fall on the same line.

你需证明的命题是那些三条外切线的交点总是落在同一条直线上。

Maybe if you're a little bit lucky,   here's one thing you might notice; every intersection point is uniquely associated  with a quadruplet of points on the exterior.

也许如果你有点幸运， 你会注意到一件事；个交点都唯一地与圆外的四个点相关联。

For example, if you calculate 4 choose 4,   everything cancels and you just get one and indeed, there is a single intersection point  in this diagram.

例如，如果你计算4选4， 所有项都会约掉， 最后得到1，确实，在这个图中只有一个交点。

You might mentally imagine turning them with a little knob to find the sweet spot angle where all those intersection points line up.

你可能会在心里想象用一个小旋钮来调整它们，找到一个所有交点都对齐的最佳角度。

If there are no objects between the intersection point and a light source, then that means that the point on the blue stripe is directly illuminated by that light source.

如果交点和光源之间没有其他物体，那么这意味着蓝色条纹上的那个点是直接被该光源照亮的。

This light is called indirect illumination, and in order to calculate it, we take the intersection point from the primary ray and generate a secondary ray that bounces off it.

这种光线被为间接照明，为了计算它， 们从主光线的交点生成一个二次光线， 该二次光线从该交点弹出。

And then similarly, if you look at the lines passing through the tips of any other pair of cones, those lines will correspond to another one of those intersection points.

然后类似地，如果你看任意另一对圆锥顶点所在的直线，这些直线将对应另一个交点。