1970 Canadian MO Problems/Problem 6
Problem
Given three non-collinear points construct a circle with centre such that the tangents from and are parallel.
Solution
Construct segment . Find the midpoint of (denote the midpoint ), by constructing its perpendicular bisector, and the point where the perpendicular bisector and meet is the midpoint of . Join to the centre of the circle, point . Construct lines parallel to the line through , and line parallel to through . From , drop a perpendicular to line , and let this point be . Construct the circle with centre and radius . Let this circle pass through E on line . Then this circle is tangent to lines and , and lines and are parallel.
Proof: We have line parallel to by our construction, and line parallel to by our construction. Thus, lines and are parallel. Thus, the line is a transversal to and . Since , by the co-interior angle theorem for parallel lines, . Thus, the circle is indeed tangent to lines and and these two lines are parallel.
1970 Canadian MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 7 |