Construction of Circumcircle of a Triangle

Key Definitions: • Circumcircle = the circle passing through all 3 vertices of a triangle • Circumcentre = centre of the circumcircle; intersection of perpendicular bisectors • Circumradius (R) = radius of the circumcircle (distance from circumcentre to any vertex)

Property: • The circumcentre is equidistant from all three vertices • For an acute triangle: circumcentre lies inside the triangle • For a right triangle: circumcentre lies on the hypotenuse (midpoint) • For an obtuse triangle: circumcentre lies outside the triangle

Tools Required: • Compass, ruler, pencil

Step-by-Step Construction:

Given: Triangle ABC (with vertices A, B, C)

Step 1: Draw the triangle • Draw triangle ABC with the given measurements using ruler and compass

Step 2: Draw perpendicular bisector of side AB • With compass, take a radius greater than half of AB • Draw arcs from A (above and below AB) • Draw arcs from B (above and below AB) with the same radius • Join the intersection points of arcs — this is the perpendicular bisector of AB

Step 3: Draw perpendicular bisector of side BC • Repeat the same process for side BC • Draw arcs from B and C, join intersection points

Step 4: Mark the Circumcentre • The point where the two perpendicular bisectors intersect = Circumcentre (O) • (The perpendicular bisector of AC will also pass through this point — verify if needed)

Step 5: Measure the Circumradius • With compass, measure OA (= OB = OC = circumradius R)

Step 6: Draw the Circumcircle • Keeping compass at O with radius OA, draw a complete circle • This circle passes through A, B, and C • This is the circumcircle of triangle ABC

Verification: • Check that OA = OB = OC (all equal to circumradius R) • The circle should pass exactly through A, B, and C

Formula for Circumradius: R = (a × b × c) / (4 × Area of triangle) where a, b, c are the three sides

Also: R = a / (2 sin A) = b / (2 sin B) = c / (2 sin C) [Sine Rule]

Difference: Circumcircle vs Incircle: