 # Inscribed and Circumscribed Circles

Videos and lessons with examples and solutions to help High School students learn how to construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

## Inscribed and Circumscribed Triangles

Construction - Circumscribe a Circle Around a Triangle
Starting with a triangle, drawing a circle around the triangle so that each vertex of the triangle is a point on the circle.
(shown for acute and obtuse triangles).

Construction - Inscribe a Circle Inside a Triangle
Using a compass and a straight edge, construct a circle that fits inside a triangle so that each side of the triangle is tangent to the circle.

Circumscribed circle (circumcircle) and Inscribed circle (incircle) of a Triangle
The circumcircle of a triangle is the unique circle determined by the three vertices of the triangle. Its center is called the circumcenter (blue point) and is the point where the (blue) perpendicular bisectors of the sides of the triangle intersect.
The incircle of a triangle is the circle inscribed in the triangle. Its center is called the incenter (green point) and is the point where the (green) bisectors of the angles of the triangle intersect.
The incenter and the circumcenter coincide if and only if the triangle is equilateral. Alter the shape of the triangle by dragging the vertices.

## Inscribed Quadrilaterals

Proving Properties of Inscribed Quadrilaterals
Prove using arc / angle relationships, that opposite angles of inscribed quadrilaterals have to be supplementary.

Circle Geometry: Cyclic Quadrilateral
(What is the relationship between the angles of a quadrilateral that is inscribed in a circle?).

Demonstrates that the opposite angles of an inscribed quadrilateral are supplementary.
Move the points on the circumference of the circle. What do you notice about the angles in the circle?

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