Orthocentre

From A draw AD perpendicular to BC. Similarly draw CE perpendicular to CA and CF perpendicular to AB.

These perpendicular lines are called altitudes.

AD, BE, CF meet at a point called orthocentre.

Circum-centre

Draw the perpendicular bisectors of the sides of the triangle.

These perpendicular bisectors meet at a point called circum-centre.

Use this centre and OA as radius, we can draw a circle passing through the vertices A, B, C.

This circle is called circum-circle (or circumscribed circle).

In-centre

AD, BE, CF are angle bisectors of angles A, B, C.

These angle bisectors meet at a point called in-centre, I.

From I draw a perpendicular line IG to BC.

Use I as centre, IG as radius, we can draw a circle, the in-circle (or inscribed circle)

 

Ex-centre

Produce the sides of the triangle A, B, C.

Draw the angle bisectors of the interior angles (in light blue) and exterior angles (in red).

These angle bisectors meets at three points outside the triangle ABC, called ex-centres.

Use these points as centres, and the altitudes to the sides of the triangle as radii, three circles can be drawn. These circle are called ex-circles (or exscribed circles).