Inscribed and Circumscribed Circles

≈ 32 min

Inscribed and Circumscribed Circles

Two circles belong to every triangle, and they are found by two different constructions.

The circumscribed circle passes through all three vertices. Its centre — the circumcentre — is equidistant from the vertices, so it is found by the perpendicular bisectors of the sides.

The inscribed circle touches all three sides from inside. Its centre — the incentre — is equidistant from the sides, so it is found by bisecting the angles. Its radius is the perpendicular distance from that centre to any side.

Worked example. The two centres are usually in different places. Only in an equilateral triangle do they coincide, because its symmetry makes "equidistant from the vertices" and "equidistant from the sides" the same condition.

Core checkpoint: Bisect sides for the circle through the corners; bisect angles for the circle touching the sides.

Which construction locates the centre of the circle that passes through all three vertices of a triangle?

The perpendicular bisectors of the sides.

The bisectors of the angles.

The medians from each vertex.

A perpendicular dropped from one vertex.

Which lines must you bisect to find the centre of the circle that touches all three sides of a triangle from inside?

After locating the incentre, how should the radius of the inscribed circle be found?

Drop a perpendicular from the centre to one side; that perpendicular distance is the radius.

Measure from the centre to the nearest vertex.

Halve the shortest side of the triangle.

Use half the distance between the two angle bisectors.

In which triangle do the incentre and circumcentre fall at the same point, and why?

An equilateral triangle — its symmetry makes 'equidistant from the vertices' and 'equidistant from the sides' the same condition.

Any right-angled triangle, because of the 90° angle.

Any isosceles triangle, because two sides are equal.

They never coincide in any triangle.