Question 54
Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B$, $ABB_A$, $ACA_C$, $ACC_A$, $BCB_C$ and $BCC_B$, so that
$$
AB = AB_A = BA_B, \quad AC = AC_A = CA_C, \quad BC = BC_B = CB_C
$$
and
$$
\angle BAB_A = \angle ABA_B = \angle CAC_A = \angle ACA_C = \angle BCB_C = \angle CBC_B = \alpha \lt 90^\circ.
$$
Prove that the perpendiculars from $A$ to $B_A C_A$, from $B$ to $A_B C_B$ and from $C$ to $A_C B_C$ are concurrent.
$$
AB = AB_A = BA_B, \quad AC = AC_A = CA_C, \quad BC = BC_B = CB_C
$$
and
$$
\angle BAB_A = \angle ABA_B = \angle CAC_A = \angle ACA_C = \angle BCB_C = \angle CBC_B = \alpha \lt 90^\circ.
$$
Prove that the perpendiculars from $A$ to $B_A C_A$, from $B$ to $A_B C_B$ and from $C$ to $A_C B_C$ are concurrent.