Let the incircle of triangle BCD have center E and touch BC and BB' at points A' and F respectively.
Let the incircle of triangle ABB' have center G and touch AB and BB' at points H and I respectively.
Let r denote the common radius.
Using Trig.
1
--- = tan(30°) = tan(EBA' + GBH)
√3
tan(EBA') + tan(GBH)
= -----------------------
1 - tan(EBA')tan(GBH)
|EA'| |GH|
------- + ------
|A'B| |HB|
= ----------------------
|EA'| |GH|
1 - ------- · ------
|A'B| |HB|
|EA'| |GH|
------- + -------------
|A'B| |AB| - |AH|
= -----------------------------
|EA'| |GH|
1 - ------- · -------------
|A'B| |AB| - |AH|
r r
--- + ---------
1 2 - r√3
= ---------------------
r r
1 - --- · ---------
1 2 - r√3
3r - r2√3
= --------------
2 - r√3 - r2
or
r2 - 2√3r + 1 = 0
and the acceptable root is
r = √3 - √2
Using Areas
Let [XYZ] denote the area of triangle XYZ.
[AA'B] = [AGB] + [GDB] + [DEB] + [EA'B]
= [AGB] + ([GIB] - [GID]) + ([EFD] + [EFB]) + [EA'B]
= [AGB] + [GIB] - ([GID] - [EFD]) + ([EFB] + [EA'B])
= [AGB] + [GHB] - (0) + 2[EA'B]
or
√3 r(2 - r√3) r
--- = r + ------------ + 2 ---
2 2 2
or
r2 - 2√3r + 1 = 0
and the acceptable root is
r = √3 - √2
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