DSE · Senior secondary

Centres of a Triangle — coordinate methods

In DSE, the four centres are usually found by coordinate geometry: mid-points, slopes and the equations of altitudes, medians or perpendicular bisectors. A short formula refresh comes first; then Euler’s line, and a page for each centre.

Euler’s line is the fact that the orthocentre H, the centroid G and the circumcentre O are collinear, with HG : GO = 2 : 1.

Formulas marked Out-of-Syllabus are useful for a quick check, but they are not required DSE technique. Learn the coordinate methods on each page first, and treat the short-cut formulas as optional extras.

0Formula refresh — mid-point, slope, point-slope, two-point 1Euler’s line — H, G, O collinear, HG : GO = 2 : 1 2Orthocentre H — altitudes in coordinates 3Centroid G — medians in coordinates 4Circumcentre O — perpendicular bisectors 5Incentre I — easy cases and A = rs