You have triangle △ A B C which is orthogonal ∠ C = 90 o with circle with center O inscribed in it. If KL is the diameter, KL∥AB, K M ⊥ A B and L N ⊥ A B. Find angle ∠ M C N.

Question

You have triangle   △  A  B  C which is orthogonal   ∠  C  =      90    o   with circle with center O inscribed in it. If KL is the diameter, KL∥AB,   K  M  ⊥  A  B and   L  N  ⊥  A  B. Find angle   ∠  M  C  N.

Isaias Archer · Accepted Answer

Step 1If r is the inradius, please note that   C  S  =  K  M  =  r  ,  C  T  =  L  N  =  r.Also note that   P  S  =  P  K and   Q  T  =  Q  L.So we have   P  C  =  P  M  ,  Q  C  =  Q  N.Step 2That makes   △  P  C  M and   △  Q  C  N isosceles triangles.As   ∠  B  Q  N  =  ∠  A  ,  ∠  B  C  N  =            ∠      A        2  And as   ∠  A  P  M  =  ∠  B  ,  ∠  A  C  M  =            ∠      B        2    ∴  ∠  M  C  N  =      45    0

Roderick Bradley · Accepted Answer

Step 1It&#039;s apparent that       |    O  M      |    =      |    O  N      |    =      |    O  C      |    (  =      2    ⋅      |    O  K      |    ) and that   ∠  M  O  N  =      90    ∘  .Step 2Then by the inscribed angle theorem,   ∠  M  C  N  =      45    ∘  .

You have triangle triangle ABC which is orthogonal angle C=90^circ with circle with center O inscribed in it. Find angle angle MCN.

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