Three Points on a Parabola

Let three points A, B, C lie on a parabola with axis OO'. Let AA', BB', CC' be parallel to the axis. Form intersections

P = AB ^ CC', Q = AC ^ BB' R = BC ^ AA'.

Let the tangents to the parabola at A and B meet at M_C, those at A and C meet at M_B, and the tangents at B and C meet at M_A.

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Prove that

1. M_A lies on PQ, M_B on PR, and M_C on QR.
2. PQ is parallel the tangent at A, PR at B, and QR at C.

Conic Sections > Parabola

• The Parabola
• Archimedes Triangle and Squaring of Parabola
• Focal Definition of Parabola
• Focal Properties of Parabola
• Geometric Construction of Roots of Quadratic Equation
• Given Parabola, Find Axis
• Graph and Roots of Quadratic Polynomial
• Greg Markowsky's Problem for Parabola
• Parabola As Envelope of Straight Lines
• Generation of parabola via Apollonius' mesh
• Parabolic Mirror, Theory
• Parabolic Mirror, Illustration
• Three Parabola Tangents
• Three Points on a Parabola
• Two Tangents to Parabola
• Parabolic Sieve of Prime Numbers
• Parabolic Reciprocity
• Parabolas Related to the Orthic Triangle

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