### Triangle Sliding on Two Straight Lines: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet may suggest several problems of a different degree of generality. At the outset, it displays a right triangle with vertices sliding along two perpendicular lines. During this motion the right angled vertex P traces a segment of a straight line as well. As one of the vertices A or B is pulled along a straight line and the other vertex gets close to the point O of intersection of the two lines, the triangle grows in size for short while. In this dynamics, too, vertex P stays on the same straight line as before. Finally, the angle between the two straight lines can be changed by moving their endpoints. This modification entails a modification of the angle at P: to retain the meaning of the problem, P is kept on the circumcircle of ΔAOB. Equivalently, angle APB is kept supplementary to AOB.

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For a proof, note that, for a given triangle, angle PAB is fixed and equal to angle POB, as both are inscribed into the same circle and subtend the same arc (they may also subtend supplementary arcs). Thus angle POB is fixed and is not dependent on the position of triangle APB. It appears that all three vertices of ΔAPB lie on three lines that concur at point O. Thus a homothety with center O preserves the shape of the triangle (i.e. its angles) and moves each of the vertices along the corresponding line. It follows that vertex P traces the same straight line regardless of the size of the triangle.

The problem, with ∠AOB = ∠APB = 90° makes regular appearances in problem solving books. One, where ∠AOB = 120° and ΔAPB is equilateral has been included in Bicycle. Both formulations successfully camouflage the utter simplicity of the configuration and provide an excellent exercise for a study of properties of inscribed angles.

### References

- J. Konhauser, D. Velleman, S. Wagon,
*Which Way Did the Bicycle Go?*, MAA, 1996, #4.

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Copyright © 1996-2018 Alexander Bogomolny