Let A, B, C be three collinear points and let P be a point outside the line through A, B, C.Using the Simson's line theorem, prove that the circumcenters of the triangles PAB, PAC, PBCand the point P lie on a circle. [Hint: Note that if two circles intersect at two points X, Y thenthe line joining the centers of the circles is the perpendicular bisector of XY. Consider thetriangle with vertices at the circumcenters. What are the projections ofP on the sides of thistriangle?]

Question
Asked Oct 15, 2019
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Let A, B, C be three collinear points and let P be a point outside the line through A, B, C.
Using the Simson's line theorem, prove that the circumcenters of the triangles PAB, PAC, PBC
and the point P lie on a circle. [Hint: Note that if two circles intersect at two points X, Y then
the line joining the centers of the circles is the perpendicular bisector of XY. Consider the
triangle with vertices at the circumcenters. What are the projections ofP on the sides of this
triangle?]
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Let A, B, C be three collinear points and let P be a point outside the line through A, B, C. Using the Simson's line theorem, prove that the circumcenters of the triangles PAB, PAC, PBC and the point P lie on a circle. [Hint: Note that if two circles intersect at two points X, Y then the line joining the centers of the circles is the perpendicular bisector of XY. Consider the triangle with vertices at the circumcenters. What are the projections ofP on the sides of this triangle?]

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Sketch the diagrams with the help of geometrical ...

C
A
A, B and C are collinear and P is a point outside the line ABC. The triangle in dark is
formed by joining the circumcenters
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C A A, B and C are collinear and P is a point outside the line ABC. The triangle in dark is formed by joining the circumcenters

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