Where is the centre?
The centre of the circle lies on the bisector of the angle between the two tangents.
A circle between two lines
Two tangents are drawn from P. The circle itself is hidden. You know only the two lines.
The circle touches both lines. Its centre must be equally far from the two lines.
Where must the centre of such a circle be?
What this lesson covers
The idea
The line joining the centre to an external point bisects the angle between the two tangents from that point, so the centre lies on the bisector of that angle.
A circle between two lines
Two tangents are drawn from P. The circle itself is hidden. You know only the two lines.
The circle touches both lines. Its centre must be equally far from the two lines.
Where must the centre of such a circle be?
- On the line that cuts the angle between the tangents into two equal parts
- On one of the two tangents
- Anywhere between the two tangents
Place the centre
Drag the centre C. The circle about C is as large as it can be without crossing the nearest tangent. Find where the circle touches both tangents.
On the angle bisector
The line joining the centre to an external point bisects the angle between the two tangents from that point, so the centre lies on the bisector of that angle: ∠OPQ = ∠OPR.
The right triangles OQP and ORP are congruent (RHS), so ∠OPQ = ∠OPR by CPCT. In the toy the circle touched both tangents only when the two distances were equal, and C was then on the dashed line that cut the angle at P into two equal parts.
Notes
The centre of a circle lies on the bisector of the angle between the two tangents from an external point P: ∠OPQ = ∠OPR.
Check yourself
A circle touches both tangents drawn from an external point P. Where does its centre lie?
Two tangents PQ and PR are drawn from P to a circle with centre O. Angle QPR = 70°. Find angle OPQ (in degrees).
Answer: 35
OP bisects angle QPR, so angle OPQ = 70° ÷ 2 = 35°.
Two tangents PQ and PR are drawn from P to a circle with centre O. Angle OPQ = 28°. Find angle QPR (in degrees).
Answer: 56
OP bisects angle QPR, so angle QPR = 2 × 28° = 56°.
Why is angle OPQ equal to angle OPR?
- On the bisector of the angle between the two tangents — correct. Yes! The centre is equally far from both tangents, so it lies on the angle bisector.
- On one of the tangents. A tangent never passes through the centre. It only touches the circle.
- At the point P. P is outside the circle. The centre is inside it.
- The right triangles OQP and ORP are congruent (RHS), so their angles at P are equal. — correct. Yes! OQ = OR, OP is common and the angles at Q and R are right angles.
- Because P is the centre of the circle.. P is the outside point. The centre is O.
- Because PQ and PR are both radii.. PQ and PR are tangents, not radii. The radii are OQ and OR.