Geometry question about bisecting anglesLet △ A B C be a triangle and let O be its circumcentre. Let D and M be points on BC such that AD is perpendicular to BC and AM bisects the angle at A. Prove that AM bisects the ∠ O A D.What theorems are helpful for solving this question?

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Geometry question about bisecting anglesLet   △  A  B  C be a triangle and let O be its circumcentre. Let D and M be points on BC such that AD is perpendicular to BC and AM bisects the angle at A. Prove that AM bisects the   ∠  O  A  D.What theorems are helpful for solving this question?

Willow Avery · Accepted Answer

Explanation:Direct angle chasing works. Let the angles of the triangle be   α  ,  β  ,  γ.What is   ∠  B  A  O?Hint: Consider Isosceles triangle BAO. What is   ∠  A  O  B?What is   ∠  D  A  C?Hint: Consider right triangle DAC. What is   ∠  A  C  D?Hence, show that   ∠  O  A  M  =      90    ∘    −  γ  −      α    2    =  ∠  M  A  D.

Colton Gregory · Accepted Answer

Step 1First we consider   △  B  A  O, here,  ∠  B  O  A  =  2.  ∠  C, since The center angle is twice the circumference angle and                    ∠        B        A        O        =        ∠        A        B        O                            =                  1          2                (        180                  °                −        ∠        B        O        A        )                                          =                  1          2                (        180                  °                −        2.        ∠        C        )                                          =        90                  °                −        ∠        C            Step 2Now we consider the   △  A  C  D, here   ∠  D  A  C  =  90      °    −  ∠  CSo,                     ∠        B        A        O        =        ∠        D        A        C                                    ⟹                ∠        B        A        M        +        ∠        M        A        O        =        ∠        M        A        D        +        ∠        C        A        M                                                  ⟹                ∠        M        A        O        =        ∠        M        A        D            , since   ∠  B  A  M  =  ∠  C  A  M as AM bisects the   ∠  A.

Let triangle ABC be a triangle and let O be its circumcentre. Let D and M be points on BC such that AD is perpendicular to BC and AM bisects the angle at A. Prove that AM bisects the angle OAD.

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