The circle inside a triangle
Construct △ABC with BC = 6 cm, AB = 5 cm and AC = 4.5 cm, and draw the circle that touches all three sides.
Construct △ABC with BC = 6.5 cm, ∠ABC = 60° and ∠ACB = 45°, and draw the circle that touches all three sides.
Construct △ABC with BC = 6 cm, ∠ABC = 90° and AB = 4.5 cm, and draw the circle that touches all three sides.
Construct △ABC with BC = 7 cm, ∠ABC = 120° and AB = 3.5 cm, and draw the circle that touches all three sides.
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. Bisect ∠ABC: an arc across it from B, two equal arcs from where it cuts the arms, and the line from B.
4. Bisect ∠ACB the same way. The two bisectors meet at I.
5. From I, draw an arc that cuts BC twice. From those two points, draw arcs of the same radius below BC that meet; the line from I to their crossing meets BC at M, at 90°.
6. With I as the center and radius IM, draw a circle: it touches all three sides.
7. This is the incircle of △ABC, radius 1.4 cm.
1. Draw BC = 6.5 cm.
2. At B, construct an angle of 60°, and at C an angle of 45°. Their arms meet at A.
3. Bisect ∠ABC: an arc across it from B, two equal arcs from where it cuts the arms, and the line from B.
4. Bisect ∠ACB the same way. The two bisectors meet at I.
5. From I, draw an arc that cuts BC twice. From those two points, draw arcs of the same radius below BC that meet; the line from I to their crossing meets BC at M, at 90°.
6. With I as the center and radius IM, draw a circle: it touches all three sides.
7. This is the incircle of △ABC, radius 1.6 cm.
1. Draw BC = 6 cm.
2. At B, construct ∠ABC = 90° and cut BA = 4.5 cm on its arm. Join AC.
3. Bisect ∠ABC: an arc across it from B, two equal arcs from where it cuts the arms, and the line from B.
4. Bisect ∠ACB the same way. The two bisectors meet at I.
5. From I, draw an arc that cuts BC twice. From those two points, draw arcs of the same radius below BC that meet; the line from I to their crossing meets BC at M, at 90°.
6. With I as the center and radius IM, draw a circle: it touches all three sides.
7. This is the incircle of △ABC, radius 1.5 cm.
1. Draw BC = 7 cm.
2. At B, construct ∠ABC = 120° and cut BA = 3.5 cm on its arm. Join AC.
3. Bisect ∠ABC: an arc across it from B, two equal arcs from where it cuts the arms, and the line from B.
4. Bisect ∠ACB the same way. The two bisectors meet at I.
5. From I, draw an arc that cuts BC twice. From those two points, draw arcs of the same radius below BC that meet; the line from I to their crossing meets BC at M, at 90°.
6. With I as the center and radius IM, draw a circle: it touches all three sides.
7. This is the incircle of △ABC, radius 1.1 cm.
Why it works
Every point on the bisector of an angle is the same distance from its two arms. I is on two bisectors, so it is the same distance from all three sides. That distance is the perpendicular IM, and the circle of that radius round I touches each side once.