Two circles meet at points X and Y. Line segment [AXB] meets one circle at A and the other at B. Line segment [CYD] meets one circle at C and the other at D. Prove that [AC] is parallel to [BD].

Question

Two circles meet at points  X and  Y. Line segment  [AXB] meets one circle at  A and the other at  B. Line segment [CYD] meets one circle at  C and the other at  D. Prove that  [AC] is parallel to  [BD].

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Identify Given Information: Identify the given information and the theorem that can be applied. In this case, we are dealing with two circles intersecting at points X and Y. The line segments [AXB] and [CYD] intersect the circles at points A, B, C, and D respectively. To prove that [AC] is parallel to [BD], we can use the concept of alternate segment theorem which states that the angle between the tangent and chord at the point of contact is equal to the angle in the alternate segment.Apply Alternate Segment Theorem: Apply the alternate segment theorem to the angles formed at points A and B. Since [AXB] is a line segment intersecting the circles at A and B, angle AXB is subtended by arc XY in one circle, and angle AYB is subtended by arc XY in the other circle. According to the alternate segment theorem, angle AXB is equal to angle ADB, and angle AYB is equal to angle ACB.Apply Theorem to Angles: Apply the alternate segment theorem to the angles formed at points C and D. Similarly, since [CYD] is a line segment intersecting the circles at C and D, angle CYD is subtended by arc XY in one circle, and angle CXD is subtended by arc XY in the other circle. According to the alternate segment theorem, angle CYD is equal to angle CAD, and angle CXD is equal to angle CBD.Combine Results for Parallel Lines: Combine the results from the alternate segment theorem to show parallel lines. From the previous steps, we have angle ADB = angle AXB and angle ACB = angle AYB. Also, angle CAD = angle CYD and angle CBD = angle CXD. Since angles AXB and AYB are the same, and angles CYD and CXD are the same, it follows that angle ADB = angle ACB and angle CAD = angle CBD. This means that angles ADB and ACB are alternate interior angles, and angles CAD and CBD are alternate interior angles.Conclude Parallel Lines: Conclude that [AC] is parallel to [BD]. If alternate interior angles are equal, then the lines are parallel by the converse of the alternate interior angle theorem. Therefore, since angle ADB = angle ACB and angle CAD = angle CBD, line segment [AC] is parallel to line segment [BD].

Two circles meet at points $\mathrm{X}$ and $\mathrm{Y}$ . Line segment $[\mathrm{AXB}]$ meets one circle at $\mathrm{A}$ and the other at $\mathrm{B}$ . Line segment [CYD] meets one circle at $C$ and the other at $D$ . Prove that $[A C]$ is parallel to $[B D]$ .

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Two circles meet at points X \mathrm{X} X and Y \mathrm{Y} Y. Line segment [AXB] [\mathrm{AXB}] [AXB] meets one circle at A \mathrm{A} A and the other at B \mathrm{B} B. Line segment [CYD] meets one circle at C C C and the other at D D D. Prove that [AC] [A C] [AC] is parallel to [BD] [B D] [BD].

Two circles meet at points $\mathrm{X}$ and $\mathrm{Y}$ . Line segment $[\mathrm{AXB}]$ meets one circle at $\mathrm{A}$ and the other at $\mathrm{B}$ . Line segment [CYD] meets one circle at $C$ and the other at $D$ . Prove that $[A C]$ is parallel to $[B D]$ .