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Michael
Frequent VisitorJoined: 19 Jan 2004Posts: 24
Location: Hong Kong

Posted: Tue Apr 05, 2005 6:27 pm Post subject: Two problems in geometry



Question 1 Given a point E inside a square ABCD. It is known that DE = a, AE = b and CE = c. If AB = t, show that t satisfies the following equation:
Question 2
Given an equilateral triangle ABC. Inside triangle ABC there is a point D such that AD = a, BD = b and CD = c, find the length of AB in terms of a, b and c.





Wilson
Frequent VisitorJoined: 20 Oct 2004Posts: 79

Posted: Wed Apr 06, 2005 4:21 pm Post subject: Re: Two problems in geometry



Michael wrote:

Question 1 Given a point E inside a square ABCD. It is known that DE = a, AE = b and CE = c. If AB = t, show that t satisfies the following equation:
Question 2
Given an equilateral triangle ABC. Inside triangle ABC there is a point D such that AD = a, BD = b and CD = c, find the length of AB in terms of a, b and c.

Question 1.
Let the angle CDE be . Given that .
By using cosine formula on triangles ADE and CDE, we have:
Question 2.
Let the angle DBC be and . We have since ABC is equilateral.
By using cosine formula on triangles ADE and CDE, we have:
Substituting (1) into (2), we have , which is a quadratic equation in . 




Michael
Frequent VisitorJoined: 19 Jan 2004Posts: 24
Location: Hong Kong

Posted: Wed Apr 06, 2005 8:14 pm Post subject:



Dear Wilson, Thanks a lot for your pretty solutions. Today I also solved the second problem by using analytic geometry. My result is like this:
,where t is the length of AB. This equation is symmetric in a , b and c as expected because the roles of a , b and c in the equilateral triangle can be interchanged.
I didn’t check whether it is the same as your result but probably they are just the same.
Besides, I wonder if there is a generalization of these two problems. Yours, Michael._________________Beauty is the first test: there is no permanent place in the world for ugly mathematics.
G.H.HARDY





