An Obtuse Triangle Relationship

Suppose that triangle ABC has

integralside lengthsa= 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 istangentto 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?

- 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.
- What next?
**Hint?** **More of a Hint****.****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. -

**PART (3)**.- How could this be proved?

Let the lengths of the sides AF and BF be on length

n. Can you solve fornin terms ofc?

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