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.

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

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 .

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