Point to Point - Futility Closet

point to point 1

Here’s a triangle, ABC, and an arbitrary point, D, in its interior. How can we prove that AD + DB < AC + CB?

The fact seems obvious, but when the problem is presented on its own, outside of a textbook or some course of study, we have no hint as to what technique to use to prove it. Construct an equation? Apply the Pythagorean theorem?

“The issue is more serious than it first appears,” write Zbigniew Michalewicz and David B. Fogel in How to Solve It (2000). “We have given this very problem to many people, including undergraduate and graduate students, and even full professors in mathematics, engineering, or computer science. Fewer than five percent of them solved this problem within an hour, many of them required several hours, and we witnessed some failures as well.”

Here’s a dismaying hint: Michalewicz and Fogel found the problem in a math text for fifth graders in the United States. What’s the answer?

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Here’s one way to do it. Extend AD to CB and call the intersection E. The sum of two sides of any triangle is longer than the third side, so AD + DB < AD + (DE + EB) [because DB < DE + EB] AE + EB < (AC + CE) + EB [because AE < AC + CE] So AD + DB < AC + CB. (Another way is to extend BD to AC, rather than extending AD to CB, and then follow a similar argument.)