Let ABC be an acute angled triangle , and let P and Q be two points on side BC.Construct point C1 in such a way that convex quadrilateral APBC1 is cyclic, QC1||CA, and C1 and Q lie on opposite sides of line AB. Construct point B1 in such a way that convex quadrilateral APCB1 is cyclic, QB1 ||BA, and B1 and Q lie on opposite sides of line AC.Prove that points B1, C1,P and Q lie on a circle.
no idea.anyone?
In the attachment, you should be able to see that the three angles marked with a bullet (Originally Posted by nh149
) are all equal. Similarly, the three angles marked with a circle (
) are all equal.
The three marked angles at Q (circle, x and bullet) add up to 180º. Also the three angles in the triangleadd up to 180º. But these angles are an x (at Q), a circle plus a star (at
) and a bullet minus a star (at
). It follows that the two angles marked with a star (
and
) must be equal. By the "angles on the same segment" theorem, or rather its converse, it follows that the points
lie on a circle.
[It then follows fairly easily from this that the pointsare collinear.]
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