This is from the UKMT Senior Challenge of 2004.

“L, M, and N are midpoints of a skeleton cube, as shown. What is the value of angle LMN?

_____A_90°_____B_105°_____C_120°_____D_135°_____E_150°”

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This is from the UKMT Senior Challenge of 2004.

“L, M, and N are midpoints of a skeleton cube, as shown. What is the value of angle LMN?

_____A_90°_____B_105°_____C_120°_____D_135°_____E_150°”

This is a fun problem from *Mathematical Quickies* (1967).

“Prove that the sum of the vectors from the center of a regular polygon of n sides to its vertices is zero.”

See the Vector Sum Problem.

I came across this problem in Alfred Posamentier’s book, but I remember I had seen it a couple of places before and had never thought to solve it. At first, it seems like magic.

In *any* convex quadrilateral (line between any two points in the quadrilateral lies entirely inside the quadrilateral) inscribe a second convex quadrilateral with its vertices on the midpoints of the sides of the first quadrilateral. Show that the inscribed quadrilateral must be a parallelogram.

See the Magic Parallelogram.

Mathnasium of Amarillo had a nice follow up on the sum of angles type of puzzle.

“Find the total value of pink colored angles.”

Contributors to his twitter post provided not only my solution but another one that was specific to this particular configuration. See More Sum of Angles.

This is a riff on a classic problem, given in *Challenging Problems in Algebra*.

“N. Bank and S. Bank are, respectively, the north and south banks of a river with a uniform width of one mile. Town A is 3 miles north of N. Bank, town B is 5 miles south of S. Bank and 15 miles east of A. If crossing at the river banks is only at right angles to the banks, find the length of the shortest path from A to B.

**Challenge**. If the rate of land travel is uniformly 8 mph, and the rowing rate on the river is 1 2/3 mph (in still water) with a west to east current of 1 1/3 mph, find the shortest time it takes to go from A to B. [The path across the river must still be perpendicular to the banks.]” See the River Crossing.

This is another problem from Futility Closet, though Futility Closet provides a “solution” of sorts. They provide a set of steps without explaining where they came from. So I thought I would fill in the gap. The problem is to find the area of an irregular polygon, none of whose sides cross one another, if we are given the coordinates of the vertices of the polygon. See Polygon Areas Problem.

I have always had a tenuous relationship with the concept of angular momentum, but recently my concerns resurfaced when I did my studies on Kepler, and in particular his “equal areas law” and Newton’s elegant geometric proof. I love the fact that a simple geometric argument, seemingly totally divorced from the physical situation, can provide an explanation for why the line from the Sun to a planet sweeps out equal areas in equal time as the planet orbits the Sun, solely under the influence of the gravitational force between them. However, modern physics books invariably cite the conservation of angular momentum as the “explanation.” I indicated before in my “Kepler’s Laws and Newton’s Laws” essay that this “explanation” irritated me. In this essay I go into detail about my reservations concerning this line of argument. See Angular Momentum.

This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler’s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See Complex Numbers – Geometric Viewpoint.

James Tanton has provided further elaborations on the polygons and the sum of perpendicular distances from interior points. Again I approached the solutions with a mix of areas and vectors. It is rather impressive to see the number of variations that can be rung on the Viviani Theorem theme. See Polygon Altitude Problems II

I found this collection of related problems by James Tanton on Twitter. Even though all these problems do not involve perpendiculars, they have a common solution approach – a sort of theme and variations idea. In a later tweet Tanton refers to a Viviani Theorem associated with these types of problems. I did not recall that theorem explicitly or by name. I also have not looked it up yet, in order to solve these problems on my own. I am guessing there is a more classical Euclidean geometry proof, but I like my vector approach for its clarity. I also throw in a bit a calculus at the end for fun. See Polygon Altitude Problems I