Let Γ be a circle with center O. Let AC be a chord of  Γ (not containing O).
Let D be a point on chord AC (not A or C). The line through D and perpendicular to AC intersects Γ in points B₁ and B₂. Let H₁ and H₂ be the orthocenters of triangles
AB₁C and AB₂C respectively.

Define the point of intersection of the bisectors of angles H₁B₁O and H₂B₂O and
prove your result.
 

Conjectures based on a Geometers Sketchpad drawing:

• The point of intersection is on the circle.
• It is independent of point D.
• It is the point where the perpendicular bisector of segment AC intersects the circle (on the same side of the center.)

Posted by Jer on 2012-12-03 13:10:40