Shape induced by the condition of angle bisector passing through a fixed pointLet A, B be two points on 2D plane. For any C∈AB―, define the set S={P;∣;∠APC=∠CPB}.What is the shape of S?

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Shape induced by the condition of angle bisector passing through a fixed pointLet A, B  be two points on 2D plane. For any C∈AB―, define the set S={P;∣;∠APC=∠CPB}.What is the shape of S?

jmroberts70pbo2 · Accepted Answer

Step 1Here is a solution using coordinate geometry. If&amp;nbsp;|AB|=k&amp;nbsp;and&amp;nbsp;d|AC||BC|=t&amp;nbsp;where&amp;nbsp;C∈AB.Using angle bisector theorem,|AP||PB|=|AC||BC|=t⇒|AP|2=t2⋅|PB|2WLOG, coordinates of A is (0,0) and of B is (k,0). If coordinates of P is (x,y),we get&amp;nbsp;∼x2+y2=t2((x−k)2+y2)Step 2If&amp;nbsp;t=1,x=k2&amp;nbsp;i.e. the locus is perpendicular bisector of AB.If&amp;nbsp;t≠1,~≤x2+y2−2kt2t2−1x=−k2t2t2−1(x−kt2t2−1)2+y2=(ktt2−1)2which is a circle with center at&amp;nbsp;(kt2t2−1,0)&amp;nbsp;and radius&amp;nbsp;kt|t2−1|.

Shape induced by the condition of angle bisector

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