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Shenghui Yang

[Notebook] What's the Angle? (A math puzzle from twitter)

Shenghui Yang, WOLFRAM

Posted 10 months ago

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What's the Angle?

@Cshearer41/Catriona loves to share beautiful hand-drawn geometric puzzles.
Here is one of them she posted on twitter. You can casually play around the simulation beneath about two tangent circles bounded by a rectangle.

twitter=ServiceConnect["Twitter"];ds=twitter["GetTweet","TweetID"1206986078005219335]

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ID	1206986078005219335
Text	What’s the angle? https://t.co/tyPivEPSsJ
Date	December 17, 2019 11:14 am
Username	Cshearer41
URL	https://twitter.com/Cshearer41/status/120698607800...

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Column[{ds["Text"],twitter["GetTweet","TweetID"1206986078005219335,"Elements""Images"]〚1〛}]

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What’s the angle? https://t.co/tyPivEPSsJ

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We start from putting two circles on x-axis and fix the left one. Let its center 1 be {1,1} and radius 1 = 1. The circle on the right hand side is determined by its radius 2. By Pythagorean Theorem and circle-circle tangents:

circle-circle tangents

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

circle-circle tangents

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circle-circle tangents

MATHWORLD TOPIC



definition



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The points of tangency

and

for the four lines tangent to two circles with centers

and

and radii

and

are given by solving the simultaneous equations

(

)·(

)0

(

)·(

)0













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we can find the center of the circle on the right hand side be 2 = { 1 + 2

2

, 2 }.
Wrap up the quantities we have found in terms of 2, we find the location of the tangent point  between two circles using the following argument:

1 -  - 2 are collinear

◼

1  : 12 = 1 : (1 + 2)

◼

Thus its expression, in vector computation, is (2 - 1)1/(1 + 2 ) + 1. Use Manipulate function to if the radius of the right hand circle affects the angle we are looking for:

Manipulate[o2={1+2*Sqrt[r],r};angle={{0,1},(o2-o1)*1/(1+r)+o1,o2+{0,r}};Graphics[{{Blue,Circle[o1,1]},{Red,Thick,Circle[o2,r]},{FaceForm[Transparent],EdgeForm[{Thick,Magenta}],Rectangle[{0,0},o2+{r,r}]},{Thick,Line[angle]},{Thick,Arrow[{{1,r},angle[[2]]}]},Text[PlanarAngle[angle]/Degree,{0.9`,r}]},AxesTrue],{r,1.,5.},Initialization(o1={1,1})]

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