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1970 Canadian MO Problems/Problem 6

Problem

Given three non-collinear points $A,B,C,$ construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.

Solution

Construct segment $AB$. Find the midpoint of $AB$ (denote the midpoint $M$), by constructing its perpendicular bisector, and the point where the perpendicular bisector and $AB$ meet is the midpoint of $AB$. Join $M$ to the centre of the circle, point $C$. Construct lines $l$ parallel to the line $MC$ through $A$, and line $k$ parallel to $MC$ through $B$. From $C$, drop a perpendicular to line $k$, and let this point be $D$. Construct the circle with centre $C$ and radius $CD$. Let this circle pass through E on line $l$. Then this circle is tangent to lines $l$ and $m$, and lines $l$ and $k$ are parallel.

Proof: We have line $l$ parallel to $CM$ by our construction, and line $k$ parallel to $CM$ by our construction. Thus, lines $l$ and $k$ are parallel. Thus, the line $CD$ is a transversal to $l$ and $k$. Since $\angle BDC = 90^{\circ}$, by the co-interior angle theorem for parallel lines, $\angle AED = 180 - \angle BDC = 180 - 90 = 90$. Thus, the circle is indeed tangent to lines $l$ and $k$ 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


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