Construct circumcircles and incircles of triangles, and construct tangents to a circle from an external point.
A circumcircle passes through all three vertices of a triangle, with centre at the intersection of perpendicular bisectors. An incircle touches all three sides, with centre at the intersection of angle bisectors.
What is the centre of a circumcircle called, and how is it found?
The circumcentre, found at the intersection of the perpendicular bisectors of the triangle's sides
What is the centre of an incircle called, and how is it found?
The incentre, found at the intersection of the triangle's angle bisectors
To construct tangents from an external point to a circle, a circle is drawn with the midpoint of the point-to-centre segment as its centre; where it intersects the original circle gives the tangent points.
Why does drawing a circle on the point-to-centre segment as diameter help construct tangents?
Because any point on that circle forms a 90° angle with the original centre and external point, satisfying the tangent's perpendicularity condition
How many tangents can be drawn from an external point to a circle?
Exactly two
1. What is the point of intersection of a triangle's perpendicular bisectors called?
2. What is the point of intersection of a triangle's angle bisectors called?
3. How many tangents can be drawn to a circle from a point on the circle itself?
4. Why must the circumcentre be equidistant from all three vertices?
5. What is used as the diameter when constructing tangents from an external point P to a circle with centre O?