# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Ceva’s Here! Topic: Geometry/Inequalities. Level: Olympiad. March 28th, 2006

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

and

as desired. QED.

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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.

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Practice Problem: (1996 USAMO – #5) Let be a triangle, and an interior point such that , , , and . Prove that the triangle is isosceles.