Jean-Marc passed on this problem to me. Here's the story. You're on one side of a pit of fire. You see two walls protruding from it. You must figure out how to build a wall so that if it were continued past the pit of fire it would bisect the angle. I solved this with trig the other day during a meeting and told Jean-Marc. He said the fun solution is purely geometric. So I'm back at work on this problem with my weak geometry. Help!
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Pick any point on the first line. Pick any point on the second line. Draw a line connecting the points. Bisect the angles formed by this line and the original lines.
It can be proved that the internal angle bisectors of any triangle intersect in a single point, and this point is the center of the inscribed circle. Let's call this point P.
The bisector of the angle on the far side of the fire pit will thus go thru P and will hit our construction line at a point. Let's call this point Q.
We need to find Q and then build our wall along the line from it to P.
Draw a line from P to either one of the original lines that is perpendicular to that line. The radius of the inscribed circle is now determined. Draw the circle. Where it intersects with our construction line is what we called Q.
Start building your wall.
A triangle's angle bisectors meet at one point, and a circle inscribed inside of a triangle has tangent points that are on the adjacent angle's bisector.
That's awesome, Lee. I'm embarrassed to say I didn't know either of these two facts. They seem so fundamental that I really should. I'm sending this over to Jean-Marc to see if this is how he did it.
I'm going to keep this fairly short. I will follow up w/ a real proof, which I believe I now have. My first "proof" is flawed (which I discovered on a run I took after posting it).
These facts about triangle angle bisectors and inscribed circles are very cool, I agree. I was reminded of them only recently when I was working out a problem I found in the Bittinger / Beecher book we use.
It turns out that the center of the inscribed circle is used in my (hopefully) valid proof but in a completely different way. The radius of the circle drawn from the center of the circle to the tangent point on the triangle is not (usually) colinear with the angle bisector line.
I cannot resist adding one more line here: If I ever apply for a full-time teaching position at Tunxis, I certainly hope Jean-Marc forgives me for this first shot at a proof. I've been reminded many times already during my tutoring sessions that checking your work and finding errors often provides the most interesting opportunities to learn how things really work.
Can't speak for Jean-Marc, but no worries. I'm learning a lot just watching you take down this problem. Can't wait to see what you come up with for the solution.
There's a guy named Roger working on all these problems we're posing. I haven't gotten him to post to the comments yet, but he's sending in his solutions to Lee via email. He came up with another solution to the "Angle of Fire". This is actually the solution that Jean-Marc showed me. Here it is.
Solution:
(1) Pick a point on each protruding wall so that when you connect the two points, the resulting line is parallel to the Pit of Fire line. [NOTE: You must be far enough away from the Pit of Fire so that you can find the intersection of the bisectors which you will be creating in step 2.]
(2) Bisect the two angles created by the new line and the protruding walls and find the point where the bisectors intersect. Mark this point A.
(3) Pick a different point on each protruding wall. Similar to step (1), when these points are connected, a line parallel to the Pit of Fire line is created. [NOTE: Again, make sure you are far enough away from the Pit of Fire.]
(4) Bisect the two angles created by the new line and the protruding walls and find the point where the bisectors intersect. Mark this point B.
(5) Draw a line connecting points A and B and continue it to the Pit of Fire. This will be the line for the wall being built.
(6) Build the wall.
I don't think it is important that the first line be parallel to the fire wall. Any two lines will do as long as they are different, and the intersection of their bisectors occurs on the same side of the wall of fire.
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