The circle through three corners
Construct △ABC with BC = 6 cm, AB = 5 cm and AC = 4.5 cm, and draw the circle through A, B and C.
Construct △ABC with BC = 5 cm, ∠ABC = 45° and ∠ACB = 60°, and draw the circle through A, B and C.
Construct △ABC with BC = 4 cm and the hypotenuse AC = 6 cm, right-angled at B, and draw the circle through A, B and C.
Construct △ABC with BC = 5 cm, ∠ABC = 120° and AB = 3 cm, and draw the circle through A, B and C.
1. Draw BC = 6 cm.
2. With B as the center and radius 5 cm, and C as the center and radius 4.5 cm, draw arcs that cut at A. Join AB and AC.
3. Draw the perpendicular bisector of BC: arcs of the same radius from B and C, above and below, and the line through their crossings.
4. Draw the perpendicular bisector of AB the same way. The two bisectors meet at O.
5. With O as the center and radius OA, draw a circle: it passes through A, B and C.
6. This is the circumcircle of △ABC, radius 3.1 cm.
1. Draw BC = 5 cm.
2. At B, construct an angle of 45°, and at C an angle of 60°. Their arms meet at A.
3. Draw the perpendicular bisector of BC: arcs of the same radius from B and C, above and below, and the line through their crossings.
4. Draw the perpendicular bisector of AB the same way. The two bisectors meet at O.
5. With O as the center and radius OA, draw a circle: it passes through A, B and C.
6. This is the circumcircle of △ABC, radius 2.6 cm.
1. Draw BC = 4 cm.
2. At B, construct an angle of 90°. With C as the center and radius 6 cm, cut its arm at A. Join AC.
3. Draw the perpendicular bisector of BC: arcs of the same radius from B and C, above and below, and the line through their crossings.
4. Draw the perpendicular bisector of AB the same way. The two bisectors meet at O.
5. With O as the center and radius OA, draw a circle: it passes through A, B and C.
6. This is the circumcircle of △ABC, radius 3 cm, with O the midpoint of the hypotenuse.
1. Draw BC = 5 cm.
2. At B, construct ∠ABC = 120° and cut BA = 3 cm on its arm. Join AC.
3. Draw the perpendicular bisector of BC: arcs of the same radius from B and C, above and below, and the line through their crossings.
4. Draw the perpendicular bisector of AB the same way. The two bisectors meet at O.
5. With O as the center and radius OA, draw a circle: it passes through A, B and C.
6. This is the circumcircle of △ABC, radius 4 cm, with O outside the obtuse triangle.
Why it works
Every point on the perpendicular bisector of BC is the same distance from B as from C, and every point on that of AB the same distance from A as from B. Where they meet, O is the same distance from all three corners, so the circle round O through one corner goes through all three.