Circles
Circle Intersections
Grade 11
Question:
<p>Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q, and intersect in M and another point N.</p><p>The point of intersection of the lines FA and BC is:</p>
<p>(a) N</p>
<p>(b) outside S₁ but inside S₂</p>
<p>(c) outside S₂ but inside S₁</p>
<p>(d) inside S₁ and S₂ both</p>
Step-by-Step Solution
Key Concept: Use properties of square circumcircles and radical axis to show that FA ∩ BC = N.
<p><strong>Analysis:</strong> By the properties of the circumcircles of the two squares and the geometric configuration, the intersection point of lines FA and BC coincides with point N, the second intersection point of circles S₁ and S₂.</p>
Correct Answer: a