Construct the circumcenter or incenter of a triangle (Tutorial Only)

Learning Objectives Define circumcenter, incenter, perpendicular bisector, and angle bisector. Differentiate between the construction methods for the circumcenter and the incenter. Accurately construct the perpendicular bisector of a line segment using a compass and straightedge. Accurately construct the angle bisector of an angle using a compass and straightedge. Construct the circumcenter of any triangle by finding the point of concurrency of the perpendicular bisectors. Construct the incenter of any triangle by finding the point of concurrency of the angle bisectors. Explain the key property of a circumcenter (equidistant from vertices) and an incenter (equidistant from sides). Imagine a city wants to build a new library that is the exact same distance from three differ...

TermDefinitionExample Perpendicular BisectorA line, segment, or ray that is perpendicular to a segment at its midpoint.For a segment AB with midpoint M, the line passing through M such that it forms a 90° angle with AB is the perpendicular bisector. Angle BisectorA ray that divides an angle into two adjacent, congruent 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 perpendicular bisectors of 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 triangle's circumscribed circle.In ΔABC, the point P where the perpendicular bisectors of sides AB, BC, and AC...

Circumcenter Theorem If P is the circumcenter of ΔABC, then PA = PB = PC = r, where r is the radius of the circumscribed circle. This theorem states that the circumcenter of a triangle is equidistant from the three vertices. This property is why it serves as the center of a circle that passes through all three vertices. Incenter Theorem If I is the incenter of ΔABC and ID ⊥ AB, IE ⊥ BC, IF ⊥ AC, then ID = IE = IF = r, where r is the radius of the inscribed circle. This theorem states that the incenter of a triangle is equidistant from the three sides. This property is why it serves as the center of a circle that is tangent to all three sides. Location of Centers Circumcenter: Acute Δ (inside), Right Δ (on hypotenuse), Obtuse Δ (outside). Incenter: Always inside the tri...