Trigonometry

Isosceles Triangle and Square of Equal Area in a Circle

An isosceles triangle and a square with equal areas are inscribed in a circle as shown. The construction was created using Geogebra, iteratively zooming in and adjusting it until everything fitted precisely. The apex angle of the triangle measures 94.6° approximately, a fact that can be used to check your own analysis. Unable to find algebraic results, I concluded that the exercise for the reader is:

Express simultaneous equations for this construction, and use computer algebra software to find numeric values for the lengths of the edges. You may define the radius of the circle as 1. Note that the base of the triangle is slightly shorter than the diameter of the circle. The centre of the circle is unmarked, locating it being part of the exercise.

An alternative exercise is to take the apex angle as given, define the height of the triangle as 1, and create an accurate construction using trigonometry and a square root calculation.

May 18, 2022 by Computer Algebra Software Geogebra Geometry Trigonometry 0

Proof that the Golden Ratio is in the 3-4-5 Right Triangle

In my previous post Golden Rectangles in a Right Triangle I inscribed circles in a right-angled triangle with sides of length 3, 4, and 5. I measured that the radius of one inscribed circle was 1/φ2. This was an empirical observation of limited accuracy, not a mathematical proof. Phi or φ, equal to (1+√5)/2, is a transcendental number. Note Paul’s correction in the comments : phi is not transcendental. Fortunately, my colleague Ed Staples was able to prove the length of the radius using trigonometry. His work is shown in the figure. The length r to be calculated is the radius of the small circle inscribed in the corner between the sides of length 3 and length 5. He proves that r = 2 – φ. And, 2 – φ is equal to 1/φ2, which is the value that was determined approximately from the construction. I used the Geogebra software to make this construction. The software conveniently measures lengths and angles, which was particularly useful in this case. You can find Ed Staples’ blog, with a variety of mathematical puzzles and tutorials, at https://www.mathematicalwhetstones.com/blog

February 20, 2022 by Geometry Trigonometry 2