@djmathman has compiled a list of interesting geometry problem on AOPS. I would like to show how to use Wolfram Language to solve them.

Problem

In the figure shown below, circle B is tangent to circle A at X, circle C is tangent to circle A at Y, and circles B and C are tangent to each other. If AB = 6, AC = 5, and BC = 9, what is AX ?

Find the Triangle ABC

In Mathematica you can handle the solution and visualization of the problem altogether very quickly. We would like to use the documented convention of the SSS triangle with A at origin and B on the positive side
In[]:=
tri=SSSTriangle[9,5,6];
Though we have a rough idea about the configuration through the given illustration, it is quite off from the actual SSS triangle. The other quick way you can check the obtuse angle at vertex A by
2
AC
+
2
BC
<
2
CA
⇔
2
5
+
2
6
<
2
9
, which implies Cos[A] is negative.
In[]:=
Graphics
,EdgeForm[{Thick,Yellow}],tri
Out[]=
Extract the vertices
In[]:=
vx=AssociationThread[{a,b,c}tri〚1〛]
Out[]=
a{0,0},b{6,0},c-
5
3
,
10
2
3


Solve for Circles

Because we know that all three circle are pairwise tangent, the distance between centers can be simply computed with addition or subtraction of radii. Given the configuration, we can find three equations to solve three unknown radii:
In[]:=
radii=Association@With[{ab=6,bc=9,ca=5},​​Solve[{rc+cara,rb+abra,rb+rcbc},{ra,rb,rc}]​​]
Out[]=
ra10,rb4,rc5

Visualize the Solution

For each circle, we have its center as a known vertex and radius as the solution from the equations above. we can easily visualize three circles and the given triangle with all dimensions fit proportionally.
In[]:=
Graphics​​Dashed,
,Thick,​​Circle[vx[a],radii[ra]],​​Circle[vx[b],radii[rb]],​​Circle[vx[c],radii[rc]],​​
,EdgeForm[{Thick,Yellow}],tri,​​Text[Style[ToUpperCase@ToString[#],18],vx[#]-0.4]&/@{a,b,c}​​
Out[]=
Eventually, we can see that AX is just
r
a
=10

Remark

◼
  • Because we have
    AC=5
    and
    r
    c
    =5
    , we can conclude that A is on the circle centered at C
    • ◼
  • Problem 001 from this series

    • CITE THIS NOTEBOOK

    Bridging the Olympiad Gap problem: radius of a circle with two inner tangent circles
    by Shenghui Yang​
    Wolfram Community, STAFF PICKS, 2020
    ​https://community.wolfram.com/groups/-/m/t/1881573