O lvl A Maths: Plane Geometry

Question from http://www.sgforums.com/forums/2297/topics/368889

In the diagram, XTPY is a tangent to 2 circles at P and ABTC is a tangent to the circle of centre O at C.
The line AP is a chord of one circle and produced to meet the other circle at Q.

(i) Explain why a circle passes through O,C, T and P

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Answer:

Let angle COP = 2x

Hence angle CQP=x (angle at centre = 2 angle at circumference)

angle PCB = angle CQP =x (alt. segment theorem)

Also, angle CPT= angle CQP=x (alt. segment theorem)

Hence, angle CTP= 180 - 2x (sum of angles in triangle)

Since angle COP + angle CTP = 180, quadrilateral CTPO is a cyclic quad, at which a circle will pass through all of the points. (shown)


Thanks to Leekeewei at sgforums for providing the solution.

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