When two straight lines pass through the center of a circle with diameter AA' and are orthogonal to each other, and two points Q and Q' on the circle’s circumference are symmetrical with respect to the straight line passing through A and A' the intersection point of the two lines AQ and A'Q' will lie on the tangent of the hyperbola at A or A' on the circle. This theorem holds as a result of the fact that the product of the slopes of AQ and A'Q' is equal to one. This geometric concept can be effectively illustrated through “proofs without words (PWW)” using the ratio of similitude of right triangles.

circle, hyperbola, ratio of similitude.

Received: August 15, 2023;  Accepted: October 3, 2023; Published: November 10, 2023

How to cite this article: Yukio Kobayashi, Exploring of the relationship between a circle and a hyperbola, Far East Journal of Mathematical Education 25 (2023), 47-55. http://dx.doi.org/10.17654/0973563123015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] Chuya Fukuda, Construction of hyperbola, Far East Journal of Mathematical Education 24 (2023), 35-36.
[2] Roger G. Nelsen, Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials), The Mathematical Association of America, 1997.