Prove that the line segment joining
the points of contact of two parallel tangents to a circle passes through the
Let’s begin by sketching this
out. Remember a diagram does not need to
be to scale. But it should be roughly in
proportion to help us decide how best to answer the question. Here, we have a circle with centre
𝑂 and two parallel tangents that meet the circle at the points 𝑃 and 𝑄. Let’s add in the lines joining the
centre of the circle 𝑂 to the point where the tangents meet the circle at 𝑃 and
By their very definition, these are
the radii of the circle. We know that the radii and the
tangent form an angle of 90 degrees. Since the two tangents are parallel
and the angle between both of the tangents and their radii is 90 degrees, this in
turn means that the two radii formed by the line segments 𝑂𝑃 and 𝑂𝑄 must be
parallel to one another.
We also know that the radii both
pass through the centre of the circle 𝑂. This means they are part of the
same line and the points 𝑃, 𝑂, and 𝑄 are collinear. We have shown that the two radii we
added in are parallel and form part of the same line. This means they actually form a
single straight line passing through the centre of the circle at 𝑂.
We have, therefore, proved that the
line segment joining the points of contact of two parallel tangents to a circle
passes through the circle centre.