This morning my older son and I worked through a great example problem in Art of Problem Solving’s Introduction to Geometry book. By amazing luck the section is one of the sections that Art of Problem Solving highlights on their web page about the book, so feel free to check out problem 5.7 here (and don’t peek at the solution!):
We actually came across the problem yesterday, but I wanted to devote an entire day to it today because the clever use of ratios in this problem is so instructive. I definitely didn’t want the mathematical beauty in this example to be lost because we had to rush through it. Also, we’ve been away from fractions and ratios for a while so my guess was that it would take a full hour to go through the problem in detail. It did.
In the middle of talking through the problem this morning I remembered that the proof of Ceva’s theorem also uses ratios in a clever way, and thought a fun follow up on the example from this morning would be walking through the proof of Ceva’s theorem tonight. It really is amazing that you can prove this beautiful theorem with just the area formula for triangles and a clever use of ratios.
In the presentation below, I’m following the proof given in section 1.2 of Geometry Revisited, which is where I learned about the theorem back in high school (and since I was too lazy to take a new picture, check out C.D. Olds’s Continued Fractions book too!!):
I started with the statement of the problem and showed how to get started on the proof by making some simple observations about areas of triangles. Then we began looking at the neat ratio idea:
Since this idea about ratios really isn’t that intuitive I wanted to take a little break from the geometry to just get a better understanding of why ratios behave in this seemingly strange way. My son had the nice idea to look at the relationship abstractly to see why it was true. It is a little funny that the relationship is easier to see abstractly than with specific numbers.
In the last part of the proof we show that the product of the three ratios is equal to one. In the first video we showed that the first ratio we were looking at was equal to the ratio of the areas of two triangles. We apply the same argument for the remaining two ratios and find two other sets of triangles whose areas are in the ratio that we were looking at originally.
If you are careful with how you label these triangles (and I wasn’t) you see quickly that the product of the three ratios is equal to one. If you aren’t careful it takes a little bit of extra time to see that all of the products cancel.
So, a really fun example. To a kid learning geometry it probably seems pretty surprising that a concept from arithmetic is going to lead to such an impressive result in geometry. I like that aspect of showing this proof, too, and I also like the reverse – namely that a day of studying geometry gave us a great chance to review ratios.
A super fun day overall, and all from one cool example from our Geometry book!
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