Epitrochoids and Triangles (and some more)

[jahz]

Yikes! I've need help with two questions that seem to involve hard and tedious trig:

1. How do I prove that an equilateral triangle can be inscribed in an epitrochoid?

2. How do I find the coordinates of a centroid of an equilateral triangle (given the x- and y- coordinates of its three vertices)?

(I have no idea where to get started)

P.S. Can anyone tell me how to find the sum of a harmonic series with all the numbers that have the digit zero removed? (E.g., (1/1 + ... 1/9 ) + (1/11 + ... 1/19)). I've gathered that I'm supposed to group the numbers as (1/1 + ... 1/9) + (1/11 + ... 1/99) + (1/111 + 1/999) + ..., but I don't know what to do from there on.

 

[vsage]

Sorry, I can't think of anything for 1 right now.

2. If I recall my calculus class right, the center of mass of a triangle of uniform density (assumed in this case) is the same as its centroid. Just average the points

For the P.S, consider the fraction of numbers you leave out by removing every number with a 0 in it from the sum. The sum of the series you mention would be (1 - that fraction) multiplied by the sum of the harmonic series (1 + 1/2 + 1/3 + 1/4...)

 

[OlderDan]

Yikes! I've need help with two questions that seem to involve hard and tedious trig:

1. How do I prove that an equilateral triangle can be inscribed in an epitrochoid?

2. How do I find the coordinates of a centroid of an equilateral triangle (given the x- and y- coordinates of its three vertices)?

(I have no idea where to get started)

P.S. Can anyone tell me how to find the sum of a harmonic series with all the numbers that have the digit zero removed? (E.g., (1/1 + ... 1/9 ) + (1/11 + ... 1/19)). I've gathered that I'm supposed to group the numbers as (1/1 + ... 1/9) + (1/11 + ... 1/99) + (1/111 + 1/999) + ..., but I don't know what to do from there on.

See the diagram here

http://mathworld.wolfram.com/Epitrochoid.html

If you make the ratio a/b = 3 the resulting figure will be closed after 3 turns of the smaller circle and it will be symmetric under rotation by 120 degrees. You should be able to take advantage of that symmetry to prove that and equilateral triangle can be inscribed.

This site will let you construct them if your browser is Java enabled

http://www-groups.dcs.st-and.ac.uk/~history/Java/Epitrochoid.html