Some people need more help than others. #iTeachMath

Given triangle ABC, we construct squares AB B1 B2 and AC C1 C2 on the sides. Let M be the midpoint of B1 and C1. As point A varies, what do you notice about the position of point M?

www.desmos.com/calculator/3...

@desmos.com #iTeachMath

Suppose we want to arrange N points on a sphere, so that the smallest distance between any two points is maximized. For N = 4, 6, 8, we can take a regular tetrahedron, regular octahedron, and cube, respectively.

One of these is not the optimal solution. Which one?

#iTeachMath

As @diffgeom.bsky.social points out, inversion takes care of the problem nicely. (And many of these Sangaku problems were solved by inversion.)

The three arcs invert to three lines. There are two circles that are tangent to these three lines; we want the "top" one.

While out on a walk, I noticed that the hubcaps on tires have very different kinds of symmetry. What kind of symmetry does your car have? What about the cars around you? #iTeachMath

Table of Lissajous Curves www.desmos.com/calculator/z...

I've seen this animation online, so I thought I'd try building it myself.

@desmos.com #iTeachMath

In the diagram below, triangle ABC is equilateral.

Given a and b, can you find p and q?

Given p and q, can you find a and b?

#iTeachMath

Three congruent circles are inscribed in a triangle, as shown below. Can you compute the common radius? www.desmos.com/calculator/p... @desmos.com #iTeachMath

This is a consequence of the Incenter-Excenter lemma (web.evanchen.cc/handouts/Fac...): In triangle ABC, if I is the incenter, I_A is the A-excenter, and M is the midpoint of arc BC, then MB = MC = MI = MI_A.

What is the sum of all the numbers in the 9 times 9 table below? #iTeachMath

We can also divide the circle into two sectors, and roll both sectors into cones. What is the maximum total volume of the two cones?

teacher.desmos.com/activitybuil...

I have been thinking about how Desmos can show multiple representations on the same graph.

For example, suppose we cut a sector out of a circle with radius 1. We then roll the sector into a cone. What is the maximum volume of the cone?

teacher.desmos.com/activitybuil...

@desmos.com #iTeachMath

Apollonian Gasket www.desmos.com/calculator/e... @desmos.com

The Simson Line www.desmos.com/calculator/x... @desmos.com

Nested Hypocycloids www.desmos.com/calculator/l... @desmos.com

Poncelet's Theorem www.desmos.com/calculator/g... @desmos.com