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?]
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?]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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?]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2707a03-8914-44d9-b3b0-567da8e5c887%2Fbe591714-8815-46b1-88a6-acf814f64434%2Fknodtyc.png&w=3840&q=75)
Transcribed Image Text: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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