Two circles intersect at points P and Q. Through points P and Q, straight lines are drawn, crossing one of the circles

| August 8, 2021 | education

Two circles intersect at points P and Q. Through points P and Q, straight lines are drawn, crossing one of the circles at points A and C, and the other at points B and D. Prove that lines AC and BD are parallel.

The quadrilateral ABQP is inscribed in a circle, then the sum of its opposite angles is 180. Let the angle ABQ = X0, then the angle APQ = (180 – X) 0.

The angle APQ is adjacent to the angle CPQ, then the angle CPQ = (180 – APQ) = (180 – 180 + X) = X0.

The quadrangle PCDQ is also inscribed in a circle, then the angle СРQ + СДQ = 180.

Angle СДQ = 180 – СРQ = 180 – Х.

The sum of the angles ABQ + SDQ = X + 180 – X = 180.

Angles ABQ and CDQ are one-sided angles at the intersection of lines AB and CD of secant AD, and since their sum is 180, lines AB and CD are parallel, which was required to be proved.



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