Let △ A B C and E, D on [ A B ] and [ A C ] s.t. BEDC is inscribable. Let P ∈ [ B D ] , Q ∈ [ C E ] , s.t. AEPC and ADQB are also inscribable. Show that A P = A Q .

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Let   △  A  B  C and E, D on   [  A  B  ] and   [  A  C  ] s.t. BEDC is inscribable. Let   P  ∈  [  B  D  ]  ,  Q  ∈  [  C  E  ] , s.t. AEPC and ADQB are also inscribable. Show that   A  P  =  A  Q .

zlepljalz2 · Accepted Answer

Step 1  ∠  A  B  D  =  ∠  E  B  D  =  ∠  E  C  D  =  ∠  A  C  E  ⇒  △  A  B  D  ∼  △  A  C  E  ⇒             A      D              A      B        =            A      E              A      C        ⇒  A  D  ⋅  A  C  =  A  B  ⋅  A  E  ∠  E  P  A  =  ∠  E  C  A  =  ∠  E  C  D  =  ∠  E  B  D  =  ∠  P  B  A  ⇒  △  E  P  A  ∼  △  P  B  A  ⇒            A      E              A      P        =            A      P              A      B        ⇒  A      P    2    =  A  B  ⋅  A  EIn the same way:   A      Q    2    =  A  C  ⋅  A  DThen   A      P    2    =  A  B  ⋅  A  E  =  A  C  ⋅  A  D  =  A      Q    2   ,   A  P  =  A  Q

rmd1228887e · Accepted Answer

Step 1  ∠  A  Q  R  ≠  ∠  A  D  P  as mentioned in comment.I made two drawings, one isosceles. These points are noticeable:The measure of the sides and angles of isosceles triangle APQ does not vary with the type of triangle.It can be assumed as a kind of affine transformation. Because base of triangle is fixed and the location of vertex A alters. It is like moving a triangle APQ relative to the base of ABC to construct three circles with certain measures of diameter.

Let <mi mathvariant="normal">△ A B C and E, D on [ A B

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