You have a quadrilateral ABCD. I want to find all the points x inside ABCD such that angle(A,x,B)=angle(C,x,D) Is there a known formula that gives these points ?

deiluefniwf 2022-09-25 Answered
You have a quadrilateral ABCD. I want to find all the points x inside ABCD such that
a n g l e ( A , x , B ) = a n g l e ( C , x , D )
Is there a known formula that gives these points ?
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Answers (1)

Klecanlh
Answered 2022-09-26 Author has 11 answers
If you understand A,B,C,D,x as complex numbers then your condition is
x A x B / x C x D R .
Let us denote that real number t, i.e. you have equation
( x A ) ( x D ) = t ( x B ) ( x C ) .
For any given t it is a quadratic equation for x, so we can solve it; the solution doesn't look very pretty:
x = ± ( A + B t + C t D ) 2 4 ( 1 t ) ( A D B C t ) A + B t + C t D 2 ( t 1 ) .
Anyway, this gives you the points you're looking for (parametrized by t R ).
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