@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.
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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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What’s the angle? https://t.co/tyPivEPSsJ
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
circle-circle tangents
MATHWORLD TOPIC

definition

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The points of tangency
t
1
and
t
2
for the four lines tangent to two circles with centers
x
1
and
x
2
and radii
r
1
and
r
2
are given by solving the simultaneous equations
(
t
2
-
x
2
)·(
t
2
-
t
1
)0
(
t
1
-
x
1
)·(
t
2
-
t
1
)0
2

t
1
-
x
1


2
r
1
2

t
2
-
x
2


2
r
2
.
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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    ​
    r
    The proof is left to you as homework. To verify your argument, you can go to Catriona's twitter page to find the result in the comment section.

    CITE THIS NOTEBOOK

    What's the angle? (A math puzzle from twitter)​
    by Shenghui Yang​
    Wolfram Community, STAFF PICKS, 2019
    ​https://community.wolfram.com/groups/-/m/t/1843528