Constructing the symmetry point if z is inside the circle. Prove that for a poin

Question

Constructing the symmetry point if z is inside the circle. Prove that for a point z inside a circle c with center z0, the following construction finds the symmetry point of z. Drow the ray from z0 through z. Construct the perpendicular to this ray at z. let T be a point of intersection of this perpendicular and c. Construct the radius z0 T. Construct the perpendicular to this radius at T. The symmetric point z* is the point of intersection of this perpendicular and the ray from z0 through z. If z inside the circle of invertion; If z is outside the circle of invertion.

Explanation / Answer

in T Z Z0 ~ Z* T Z0 (similar Triangles)

Thus, TZ0/Z0Z = Z*Z0/TZ0

therefore, TZ0 x TZ0 = Z*Z0 x Z0Z ( 2 = Z*Z0 x Z0Z )

Hence, Z*Z0 & Z0Z are inverse points of each other.

Hence Proved.

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