Construct the inscribed or circumscribed circle of a triangle (Tutorial Only)

Learning Objectives Define the circumcenter and incenter of a triangle and describe their key properties. Accurately construct the perpendicular bisectors of a triangle's sides to locate the circumcenter. Accurately construct the angle bisectors of a triangle's angles to locate the incenter. Use a compass and straightedge to construct the circumscribed circle (circumcircle) for any given triangle. Use a compass and straightedge to construct the inscribed circle (incircle) for any given triangle. Differentiate between the construction methods for the inscribed and circumscribed circles. Predict the location of the circumcenter based on the type of triangle (acute, right, or obtuse). How could a city planner find the perfect spot for a new park so that it's th...

TermDefinitionExample Perpendicular BisectorA line, segment, or ray that is perpendicular to a segment at its midpoint.For a line segment AB, its perpendicular bisector passes through the middle point of AB and forms a 90° angle with AB. Angle BisectorA ray that divides an angle into two congruent, adjacent angles.If ray BX bisects ∠ABC, then m∠ABX = m∠XBC. Point of ConcurrencyThe point where three or more lines, rays, or segments intersect.The intersection point of the three angle bisectors in a triangle is a point of concurrency. CircumcenterThe point of concurrency of the three perpendicular bisectors of the sides of a triangle. It is the center of the circumscribed circle.The circumcenter is equidistant from all three vertices of the triangle. For ΔABC, if P is the circumcenter, then...

Circumcenter Theorem The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter, which is equidistant from the vertices of the triangle. If P is the circumcenter of ΔABC, then PA = PB = PC = R, where R is the radius of the circumcircle. This theorem is the foundation for constructing a circumscribed circle. By finding the intersection of the perpendicular bisectors, you find the center of the circle that will perfectly pass through all three triangle vertices. Incenter Theorem The angle bisectors of a triangle intersect at a point called the incenter, which is equidistant from the sides of the triangle. If I is the incenter of ΔABC, then the perpendicular distance from I to each side is equal to r, the radius of the incircle. This the...