Hinge TheoremDate: 12/12/2000 at 18:54:06 From: Gina Subject: Proofs in geometry How would you write a proof for the Hinge theorem? Date: 12/13/2000 at 05:10:36 From: Doctor Floor Subject: Re: Proofs in geometry Hi, Gina, Thanks for writing. Hinge Theorem: If of triangle ABC and A'B'C' sides AB = A'B' and AC = A'C', and the included angle at A is larger than the included angle at A*, then BC > B'C'. Proof: A A' /|\ /| / | \ / | / | \ / | / | \ B'/ | B | X \C |C' D We construct AD such that AD = A'C' = AC and angle BAD = angle B'A'C'. Triangles ABD and A'B'C' are congruent. Therefore BD = B'C'. Let X be the point where the angle bisector of angle DAC meets BC. From the congruent triangles AXC and AXD (SAS) we have that XD = XC. Now, by the triangle inequality we have that BX + XD > BD, so BX + XC > BD, and consequently BC > BD = B'C'. If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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