Difference between revisions of "PaperMath’s circles"
(→PaperMath’s circles) |
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<asy> | <asy> | ||
size(10cm); //Asymptote by PaperMath | size(10cm); //Asymptote by PaperMath | ||
− | |||
− | |||
real s = 0.218; | real s = 0.218; | ||
pair A, B, C, D, E; | pair A, B, C, D, E; | ||
− | |||
− | |||
A = dir(90 + 0*72)*s/cos(36); | A = dir(90 + 0*72)*s/cos(36); | ||
B = dir(90 + 1*72)*s/cos(36); | B = dir(90 + 1*72)*s/cos(36); | ||
Line 17: | Line 13: | ||
D = dir(90 + 3*72)*s/cos(36); | D = dir(90 + 3*72)*s/cos(36); | ||
E = dir(90 + 4*72)*s/cos(36); | E = dir(90 + 4*72)*s/cos(36); | ||
− | |||
− | |||
draw(A--B--C--D--E--cycle); | draw(A--B--C--D--E--cycle); | ||
− | |||
real r = 1; // Radius of the congruent circles is 1 unit | real r = 1; // Radius of the congruent circles is 1 unit | ||
draw(circle(A, r)); | draw(circle(A, r)); | ||
Line 27: | Line 20: | ||
draw(circle(D, r)); | draw(circle(D, r)); | ||
draw(circle(E, r)); | draw(circle(E, r)); | ||
− | |||
− | |||
pair P_center = (A + B + C + D + E) / 5; | pair P_center = (A + B + C + D + E) / 5; | ||
+ | real R_central = 1/cos(pi/180*54) - 1; | ||
+ | draw(circle(P_center, R_central)); | ||
+ | </asy> | ||
+ | Here is a diagram of what <math>n=8</math> would look like. | ||
− | real R_central = 1/cos(pi/180* | + | <asy> |
+ | size(10cm); // Asymptote by PaperMath | ||
+ | real s = 2.28; | ||
+ | pair A, B, C, D, E, F, G, H; | ||
+ | A = dir(90 + 0*45)*s/cos(22.5); | ||
+ | B = dir(90 + 1*45)*s/cos(22.5); | ||
+ | C = dir(90 + 2*45)*s/cos(22.5); | ||
+ | D = dir(90 + 3*45)*s/cos(22.5); | ||
+ | E = dir(90 + 4*45)*s/cos(22.5); | ||
+ | F = dir(90 + 5*45)*s/cos(22.5); | ||
+ | G = dir(90 + 6*45)*s/cos(22.5); | ||
+ | H = dir(90 + 7*45)*s/cos(22.5); | ||
+ | draw(A--B--C--D--E--F--G--H--cycle); | ||
+ | real r = 1; // Radius of the congruent circles is 1 unit | ||
+ | draw(circle(A, r)); | ||
+ | draw(circle(B, r)); | ||
+ | draw(circle(C, r)); | ||
+ | draw(circle(D, r)); | ||
+ | draw(circle(E, r)); | ||
+ | draw(circle(F, r)); | ||
+ | draw(circle(G, r)); | ||
+ | draw(circle(H, r)); | ||
+ | pair P_center = (A + B + C + D + E + F + G + H) / 8; | ||
+ | real R_central = 1/cos(pi/180*67.5) - 1; // Updated radius of the central circle | ||
draw(circle(P_center, R_central)); | draw(circle(P_center, R_central)); | ||
</asy> | </asy> |
Revision as of 20:29, 27 March 2024
Contents
PaperMath’s circles
This theorem states that for a tangent externally tangent circles with equal radii in the shape of a
-gon, the radius of the circle that is externally tangent to all the other circles can be written as
and the radius of the circle that is internally tangent to all the other circles can be written as
Where
is the radius of one of the congruent circles and where
is the number of tangent circles. The formula for the radius of the externally tangent circle is true for all values of
, since there would obviously be no circle that could be drawn internally tangent to the other circles at
Here is a diagram of what would look like.
Here is a diagram of what would look like.
Proof
We can let be the radius of one of the congruent circles, and let
be the radius of the externally tangent circle, which means the side length of the
-gon is
. We can draw an apothem of the
-gon, which bisects the side length, forming a right triangle. The length of the base is half of
, or
, and the hypotenuse is
. The angle adjacent to the base is half of an angle of a regular
-gon. We know the angle of a regular
-gon to be
, so half of that would be
. Let
for simplicity. We now have
, or
. Multiply both sides by
and we get
, and then a bit of manipulation later you get that
, or when you plug in
, you get
. Add
to find the radius of the internally tangent circle to get
, and we are done.
Notes
PaperMath’s circles was discovered by the aops user PaperMath, as the name implies.