Here is another engaging problem from Presh Talwalkar.
“___________Triangle Area 1984 AIME
Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”
See the Pinwheel Area Problem
This is another fairly simple puzzle from Futility Closet.
“If an equilateral triangle is inscribed in a circle, then the distance from any point on the circle to the triangle’s farthest vertex is equal to the sum of its distances to the two nearer vertices (q = p + r).
(A corollary of Ptolemy’s theorem.)”
See A Tidy Theorem
Here is another simply amazing problem from Five Hundred Mathematical Challenges:
“Problem 154. Show that three solutions, (x1,.y1), (x2,.y2), (x3, y3), of the four solutions of the simultaneous equations
____________(x – h)² + (y – k)² = 4(h² + k²)
______________________xy = hk
are vertices of an equilateral triangle. Give a geometrical interpretation.”
Again, I don’t see how anyone could have discovered this property involving a circle, a hyperbola, and an equilateral triangle. It seems plausible when h.=.k, but it is not at all obvious for h.≠.k. For some reason, I had difficulty getting a start on a solution, until the obvious approach dawned on me. I don’t know why it took me so long.
See the Amazing Triangle Problem.
This is truly an amazing result from Five Hundred Mathematical Challenges.
“Problem 119. Two unequal regular hexagons ABCDEF and CGHJKL touch each other at C and are so situated that F, C, and J are collinear.
(i) the circumcircle of BCG bisects FJ (at O say);
(ii) ΔBOG is equilateral.”
I wonder how anyone ever discovered this.
See the Magic Hexagons
Presh Talwalkar had another interesting problem.
“A triangle is drawn inside a square with sides 4, 3, and 5, as shown. What is the length of the square’s side?”
The problem looks simple at first, but it takes some care to avoid some hideous quartic equations.
See Tipsy 3-4-5 Triangle
This is a stimulating problem from the UKMT Senior Math Challenge for 2017. The additional problem “for investigation” is particularly challenging. (I have edited the problem slightly for clarity.)
“The parabola with equation y = x² is reflected about the line with equation y = x + 2. Which of the following is the equation of the reflected parabola?
A_x = y² + 4y + 2_____B_x = y² + 4y – 2_____C_x = y² – 4y + 2
D_x = y² – 4y – 2_____E_x = y² + 2
For investigation: Find the coordinates of the point that is obtained when the point with coordinates (x, y) is reflected about the line with equation y = mx + b.”
See Flipping Parabolas.
When our daughter-in-law made wheat shocks as center-pieces for hers and our son’s fall-themed wedding reception, I naturally could not help pointing out the age-old observation that they represented a hyperboloid of one sheet. This was naturally greeted with the usual groans, but the thought stayed with me as I realized I had never proved this mathematically to myself. And so I did.
See the Hyperboloid as Ruled Surface.
(Updates 10/9/2020, 9/19/2022) Spinning Rod Demo, Spinning Umbrella
A number of recent puzzles have involved perspective views of objects. I had never really explored the idea of a perspective map in detail. So some of the properties associated with it always seemed a bit vague to me. I decided I would derive the mathematical equations for the perspective or projective map and see how its properties fell out from the equations. With this information in hand I then addressed some questions I had about the article “Dürer: Disguise, Distance, Disagreements, and Diagonals!” by Annalisa Crannell, Marc Frantz, and Fumiko Futamura concerning a controversy over Albrecht Dürer’s woodcut St. Jerome in His Study (1514). And finally, I read somewhere that a parabola under a perspective map becomes an ellipse, so I was able to show that as well. See the Perspective Map.
(Update 7/1/2019) Continue reading
I had been exploring how Kepler originally discovered his first two laws and became fascinated by what he did in his Astronomia Nova (1609), as presented by a number of researchers. Among the writers was A. E. L. Davis. She mentioned that the characterization of the ellipse that Kepler was using was the idea of a “compressed circle,” that is, a circle all of whose points were shrunk vertically by a constant amount towards a fixed diameter of the circle. I did not recall ever hearing this idea before and tried to track down its origin together with a proof — futilely, Davis’s references notwithstanding. I then tried to prove it myself. It was easy to do with analytic geometry. But in the spirit of the Kepler era (before the advent of Fermat’s and Descartes’s beginnings at fusing algebra and geometry) I tried to prove it solely within Euclid’s plane geometry. Some critical steps seemed to come from the great work of Apollonius of Perga (262-190 BC) on Conics. But for me a final elegant proof was not evident until 1822 when Dandelin employed his inscribed spheres. See Kepler’s Ellipse.
In the process of exploring the compressed circle idea I acquired an immense appreciation and regard for Kepler and his perseverance in the face of the dominant paradigm of his era, namely, the 2000 year old idea that the celestial motions were all based on the most perfect motion of all, that of circles. The kinds of extremely laborious calculations he went through (just prior to the invention of logarithms by John Napier) were daunting, especially considering the trials he was undergoing in his personal life (trying to survive the religious destruction between Catholics and Protestants, along with defending his mother against charges of witchcraft).
Years ago (1963) I got the paperback The Calculus:A Genetic Approach, by Otto Toeplitz, which presented the basic ideas of the differential and integral calculus from a historical point of view. One thing Toeplitz did at the end of his book that I had not seen in other texts was to show the equivalence of Kepler’s Laws and Newton’s Law of Gravity. (Since 1963 David Bressoud has developed this theme in his excellent 1991 text.) I thought I would try to emulate Toeplitz’s approach with more modern notation (vectors) and arguments in hopes of extracting the essential ideas from the clutter.
A by-product of this effort was to reveal strongly the different paths that physics and mathematics follow in understanding physical reality. The mystery is that the mathematics ends up describing the physics so well. I will return to this theme a number of times in other posts. See Kepler’s Laws and Newton’s Laws.