Circle and Tangent Constructions

Overview

Circles and tangents are integral to many geometric constructions. This note outlines the steps to construct the circumcentre and circumcircle, the incentre and incircle, and a tangent to a circle at a given point using a compass and straight edge.

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Circumcentre and Circumcircle of a Triangle

Objective: To find the circumcentre of a triangle and construct the circumcircle that passes through all three vertices of the triangle.

Method

1. Draw the triangle $\triangle ABC$
2. Construct the perpendicular bisector of each side of the triangle:

• For each side, draw arcs above and below the line with the compass radius greater than half the side's length.
• Connect the intersections to form the bisector.

3. The point where all three perpendicular bisectors intersect is the circumcentre.
4. Set the compass at the circumcentre and adjust its radius to the distance between the circumcentre and any vertex of the triangle.
5. Draw the circumcircle, which will pass through all three vertices. Result: The circumcircle is constructed, and its centre is the circumcentre.

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Incentre and Incircle of a Triangle

Objective: To find the incentre of a triangle and construct the incircle that is tangent to all three sides of the triangle.

Method

1. Draw the triangle $\triangle ABC$
2. Construct the angle bisector of each interior angle of the triangle:

• For each angle, use the compass to mark arcs intersecting the angle's arms and then construct the angle bisector by joining the intersection points of these arcs.

3. The point where all three angle bisectors intersect is the incentre.
4. Set the compass at the incentre and adjust its radius to the perpendicular distance from the incentre to any side of the triangle.
5. Draw the incircle, which will be tangent to all three sides of the triangle. Result: The incircle is constructed, and its centre is the incentre.

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Tangent to a Given Circle at a Given Point on the Circle

Objective: To construct a tangent to a circle that touches the circle at a specified point.

Method

1. Draw the circle and mark the given point $P$ on the circle.
2. Draw a radius from the centre of the circle $O$ to the point $P$
3. At point $P$, construct a perpendicular line to the radius using a compass and straight edge:

• Place the compass at $P$ and draw arcs on either side of $O$ along the radius.
• Using these arc points, construct a perpendicular line through $P$

4. Extend this line, which is the tangent to the circle at the point $P$ Result: The tangent to the circle is constructed at the given point.

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Summary

1. Circumcentre and Circumcircle:

• Construct the perpendicular bisectors of the sides of a triangle to find the circumcentre.
• The circumcircle passes through all three vertices.

2. Incentre and Incircle:

• Construct the angle bisectors of a triangle to find the incentre.
• The incircle is tangent to all three sides.

3. Tangent to a Circle:

• A tangent at a given point is perpendicular to the radius drawn to that point. These constructions are vital for understanding the relationships between circles, tangents, and triangles in geometry.