0.4<(r/h) <0.5.

Textbook formulation:

Let ABC be a rectangular triangle (∠BAC = 90°), r the radius of the inscribed circle and h the altitude attached to A. Prove that

0.4<(r/h) <0.5.

Input to the system:

Proposition["Cad_61_32",  
      any[A, B, P, M,  O, M1, N1, C, D], 
      point[ A, B] ∧ midpoint[P, A, B] ∧ 
          pon[ M, segment[A, P]] ∧ 
          pon[ O, tline[ M, M, A]] ∧ 
          eqdist[O, M, M, A] ∧ 
          inter[M1, line[ A, B], 
            tline[ O, O, B]] ∧ 
          divpoint[N1, M1, O, 2] ∧ 
          inter[C, line[ B, N1], 
            tline[ A, A, B]] ∧ 
          inter[D, line[ B, C], 
            tline[ A, B, C]]
 \[Implies] 
        inequation[4/25 <= seglengthsq[O, M]/seglengthsq[A, D]] ∧ 
          inequation[
            seglengthsq[O, M]/seglengthsq[A, D] ≤ 1/4]]

End of output

Simplify[Proposition["Cad_61_32"], 
      by -> GraphicSimplifier]

Output generated automatically by the system

End of output

Input to the system:

Prove[Proposition["Cad_61_32"], by -> GeometryProver, 
      ProverOptions -> {Method -> "AreaCAD"}]

Inserted notebook generated automatically by the system

End of inserted notebook