User talk:Jhoshen1/Sandbox

Construct triangle $\triangle AYX$ such that $\triangle AYX = \triangle AYX $.

Construct isosceles triangle $XZY''$ such that


 * $ XZ = XY = XY  $
 * $ \angle XZY = \angle XYZ = 60 \degrees + \gamma $
 * $ \angle ZXY = 180 \degrees - 2 \angle XYZ = 180 \degrees - 2 (60 \degrees + \gamma) = 60 \degrees - 2 \gamma$

Construct $\angle BXZ$ such that $\angle BXZ = \angle BXZ$
 * $\therefore \triangle BXZ \cong \triangle BXZ \;\;\;\;\;$  Angle-Side-Angle
 * $\leadsto \angle XBZ \cong \triangle XBY = \beta $
 * and
 * $BX = BX $
 * $ \angle AXB = 180 \degrees - \angle XAY - \angle XBZ'' = 180 \degrees -\alpha -\beta = \angle A'X'B'$
 * $ A'X' = AX = AX $
 * $ B'X' = BX = BX $

$\therefore \triangle B'X'Z' \cong \triangle BXZ'' \;\;\;\;\;$  Angle-Side-Angle
 * $\angle X'B'Z' =\angle XBZ'' =\beta $
 * $\angle X'A'Y' =\angle XAY'' =\alpha $