How to Make Geometry Proofs Shorter
So you know how to write a geometry proof, but you have a burning desire to know how to make a geometry proof shorter. What, are you trying to play a mathematics version of the Six Degrees of Kevin Bacon? Learning how to make geometric proofs shorter requires you to have a encyclopedic understanding of geometric theorems, plus a good feel for how the theorems apply to proofs.
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Instructions
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Write a geometry theorem. You want one that has several steps, not something obvious. Alternatively you can find one somebody else has done and make them feel bad when you turn their clunky 16 step piece of junk into a lean 5 step mathematical miracle. For the sake of illustration, use the following proof: Given an equilateral triangle ABC, prove that a line DE that bisects two sides of the triangle must be parallel to the remaining side.
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Examine the reasoning behind each step of the proof. In the example, if a line bisects side AB and bisects side BC, one proof might involve first proving that a straight line drawn from the point F to the side AC makes a right angle with both AC and line DE. This proof would take a lot of steps.
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Consider alternative ways to prove the problem. The usual culprit for long proofs is a poor starting point, like trying to hook up Tom Hanks with Kevin Bacon by starting with Big instead of Apollo 13. In the example, the poor starting point is getting hung up on the right angle business. Right angles require more proof than merely equal angles.
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Apply the short cuts to the existing solution. In the example, if a line bisects side AB and bisects side BC, one shortened proof might involve first proving that the triangle FBG is also an equilateral triangle, and then using the equal angles A and F to prove that the line AB is crossing two parallel lines, thus proving the problem.
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Check your work and make sure that your shortcuts actually work, and eliminate steps. Let's see, Tom Hanks starred in Big with Elizabeth Perkins, who starred in He Said She Said with Kevin Bacon. Tom Hanks starred in Apollo 13 with Kevin Bacon. Check.
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Tips & Warnings
Have a geometry textbook handy to check your solutions. You might also want a list of geometry theorems to give your problem solving abilities a boost.
