Prove that the locus of the incenter of the is an ellipse of eccentricity
Let S and S′ be the foci of an ellipse whose eccentricity is e.P is a variable point on the ellipse.Prove that the locus of the incenter of the is an ellipse of eccentricity .
Let P be . Let the incenter of the triangle PSS′ be (h,k). The formula for the incenter of a triangle whose side lengths are a,b,c and whose vertices have coordinates , , and is
but I could not find the relationship between h and k, whence I could not find the eccentricity of this ellipse.
Answer & Explanation
Consider an ellipse with semimajor axis a and semiminor axis b centered at (0,0), where . The eccentricity e of this ellipse is given by . Without loss of generality, let and be the foci of this ellipse, where . Note that for every point P on the ellipse. Thus, if is the incenter of for , then, as you have found out,
it can be easily seen that and . Therefore,
Let and . Then,
Therefore, the locus of the incenter of PSS′ is indeed an ellipse with the semimajor axis A and the semiminor axis B. If E is its eccentricity, then
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