The Centres of a Triangle

A triangle has more than corners and sides. Four different geometric rules locate four special points, and the surprise is that each rule makes three independently defined lines meet. The four destinations are the centroid G, the circumcentre O, the incentre I, and the orthocentre H. Their rules are different, so their dots occupy different places in the same triangle. The first rule begins at the midpoint of every side. Those three landing places are A prime, B prime, and C prime. A segment from a corner to the opposite midpoint is a median. The median from A lands at A prime, and the median from B lands at B prime. Those two meet. The third median, from C to C prime, passes through the very same point. That common point is the centroid G. The centroid is the triangle's balance point. Along the median from A, the length A G is two thirds of the whole median, while G A prime is one third. The two lengths therefore have ratio two to one. A second rule starts from the same midpoints, but turns through a right angle. A perpendicular bisector crosses a side square on and divides it into equal halves. Choose any point on that line. Its distance to B equals its distance to C, the two ends of the side it bisects. The other two perpendicular bisectors carry the same promise for their sides. All three pass through one point, the circumcentre O. Because O lies on all three bisectors, O A, O B, and O C are equal. Call their common length R. A circle with centre O and radius R therefore passes through A, B, and C. This is the circumcircle, and the gap from O back to G confirms that the circumcentre and centroid are different points. The third rule belongs to the corners. An angle bisector splits one corner into two equal angles. The angle bisectors from A and C meet the first at one point. This is the incentre I. The shortest distance from I to side A B is a perpendicular drop. The corresponding drops to B C and C A have exactly the same length. Their matching marks record the equality, and that common length is r. The circle centred at I with radius r reaches every side and crosses none. It touches A B, B C, and C A at the three perpendicular feet. This is the incircle. The fourth rule returns to perpendicular lines. An altitude passes through a corner and meets the opposite side at a right angle. The altitudes from B and C pass through the first altitude at one point. This fourth meeting point is the orthocentre H. For this acute triangle, every altitude foot and the orthocentre lie inside. An obtuse angle changes that geometry.

At R, the angle is greater than a right angle, so this triangle is obtuse. The question is still the same: where do its three altitudes meet? Continue side Q R beyond R. The perpendicular from P meets that extended line outside the triangle. Continue side R P beyond R as well. The perpendicular from Q has its own right-angle foot beyond the triangle. The altitude from R meets P Q on the side itself. Its right-angle mark is inside, but the line continues upward to meet the other two altitudes. All three altitudes still meet at one point. The orthocentre H lies outside an obtuse triangle. The defining rule did not fail. Extending the two short sides exposed the same concurrency beyond the triangle.

The four centres return on a triangle whose top corner can move. First, the three side midpoints mark where the medians land. The median from A reaches A prime, the median from B reaches B prime, and the median from C reaches C prime. Their meeting point is the centroid G. The perpendicular bisectors meet at the circumcentre O, the centre of the circle through all three corners. The altitudes meet at the orthocentre H. The construction lines can now leave. Only three labelled points remain: the circumcentre O, the centroid G, and the orthocentre H. They lie on one straight line, exactly. This is the Euler line. That alignment belongs to the triangle, not to this one shape. As the top corner moves left, O, G, and H move with it and remain collinear. Move the corner back to the right, and the same line carries all three centres again. Their spacing is fixed as well. O G is one part, while G H is two parts, so O G to G H is one to two. The incentre I also moves with the triangle, but it does not generally lie on the Euler line. The centroid G comes from medians, the circumcentre O from perpendicular bisectors, the incentre I from angle bisectors, and the orthocentre H from altitudes. Four independent rules reveal a remarkably organised triangle.