Problem: (1996 IMO Shortlist – #13) Let be an equilateral triangle and let be a point in its interior. Let the lines meet the sides at the points , respectively. Prove that
Solution: First thing to take into account is the fact that the triangle is equilateral. There is probably a good reason for this, so let’s try and find it. Applying the Law of Cosines on triangle , we find
But since , we have
Similarly, and . Multiplying these three inequalities together, we obtain
But by Ceva’s, we know , so
as desired. QED.
Comment: After we were able to figure out an effective way to use the equilateral condition, it wasn’t hard to see the inequality there. Then the problem just asks for Ceva, which fits in nicely at the end.
Practice Problem: (1996 USAMO – #5) Let be a triangle, and an interior point such that , , , and . Prove that the triangle is isosceles.
Leave a Reply
You must be logged in to post a comment.