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An Obtuse Triangle Relationship

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Suppose that triangle ABC has integral side lengths a = BC, b = CA, c = AB and side AB is the longest side. Construct a square ABDE on the side of AB remote from C. Suppose furthermore that side DE of the square ABDE is tangent to the circumcircle of triangle ABC.

(1) Express c as a function of a and b.

 

(2) Find (non-similar) triangles which satisfy the hypotheses of the problem.

 

(3) Is the set of non-similar triangles which satisfy the hypotheses of the problem finite or infinite? Why?

 

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Reference: This is problem SSM 4697, Proposed by V. C. Bailey, Naples, Florida, in the January 1999 issue of School Science and Mathematics

Click here for a GSP Sketch. Move point C to different spots on the minor arc AB to view different triangles with this configuration.

 

Strategy: Finding an expression for c in terms of a and b can ignore the requirement for integer values. Once we have an expression for (1) we can use it to search for integer values of a, b, and c.

 

Must the point of tangency of the circumcircle to DE be at the midpoint? Why?

 

Draw in segments FA, FC, and FB.

Let the lengths of the sides AF and BF be on length n. Can you solve for n in terms of c?

 

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What next? Hint?

 

More of a Hint.

Solution to Part (1)

 

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PART (2)

 

When we have c as function of a and b, whether or not the values are integer, then Part (2) becomes a search for integer values to satisfy the function. Since 5 and 6 are relatively prime the search for integer values of a and b where the expression under the radical is a perfect square can be limited to where at least one of them is a multiple of 5. For example, a = 9, b = 10, yields c = 17.

 

Find others. Click here to see a short list of others.

 

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PART (3).

How could this be proved?