Let ABC be a triangle D a point on BC and r_1,r_2 the inradii of ABD, ACD resp. To construct ABC if we know the angles of ABC and are given the r_1,r_2.
Let AE = h_a be the altitude from A and r the inradius of ABC.
h_a = 2r_1r_2 / (r_1 + r_2 - r)
(by this Theorem)
r / h_a = 4Rsin(A/2)sin(B/2)sin(C/2)/2RsinBsinC = sin(A/2)/2cos(B/2)cos(C/2)
==> h_a is known.
Synthetic solution by Nikolaos Dergiades:
Hyacinthos, Message 19725