Show that MN and MP are angle bisectorsA △ A B C is drawn ( ∠ C = 90 ∘ ), in which CL ( L ∈ A B ) is bisector. The circle k with diameter CL intersects AB, BC and CA, respectively, in M, N and P. Show that MN and MP are angle bisectors of ∠ B M C and ∠ A M C.

Question

Show that MN and MP are angle bisectorsA   △  A  B  C is drawn (  ∠  C  =      90    ∘  ), in which CL   (  L  ∈  A  B  ) is bisector. The circle k with diameter CL intersects AB, BC and CA, respectively, in M, N and P. Show that MN and MP are angle bisectors of   ∠  B  M  C and   ∠  A  M  C.

canhaulatlt · Accepted Answer

Step 1  ∠  C  M  B  =  ∠  C  M  A  =      90    ∘  , since   ∠  C  M  L subtends the diameter CL.By the inscribed angle theorem,   ∠  L  P  C  =      90    ∘    =  ∠  L  N  CTriangles LPC and LNC are equal (by common side and equal angles), so   C  P  =  C  N and CPN is an isosceles right triangle.Step 2By the inscribed angle theorem,   ∠  C  M  P  =  ∠  C  N  P  =      45    ∘  , so MP is the angle bisector of   ∠  C  M  A.Analogically, MN is the angle bisector of   ∠  C  M  B.

A triangle ABC is drawn (angle C=90^circ), in which CL (L in AB) is bisector. The circle k with diameter CL intersects AB, BC and CA, respectively, in M, N and P. Show that MN and MP are angle bisectors of angle BMC and angle AMC.

Answered question

Answer & Explanation

New Questions in High school geometry