Curriculum placement
Geometry. This describes the skill, not a learner estimate.
The incircle and the circumcircle
Construct triangle incircles and circumcircles and prove cyclic angle properties
Problem types
- Find circles tied to a triangle or quadrilateral
- The centroid divides each median 2 to 1
Builds on
- Theorems about circles
- Constructions with compass and straightedge
- Regular polygons inscribed in a circle · Earlier idea
The circle through the vertices of a regular inscribed polygon is its circumscribed circle, the circle whose center is where the perpendicular bisectors of the sides meet.
- The laws of sines and cosines · Useful idea
In any triangle the law of sines gives a over sin A equal to the diameter of the circumscribed circle.
Leads to
- Equations of circles · See it applied
The circle through three points has its center where the perpendicular bisectors meet, and its equation uses that center and radius.
Related skills
- A parabola as points equally far from a focus and a line · Related idea
The circumcenter is the locus of points equally far from the three vertices, as a parabola is the locus of points equally far from a focus and a line.
- A tangent from a point outside a circle · Related idea
The incircle of a triangle is tangent to all three sides, and each radius to a point of tangency is perpendicular to that side.
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