Triangle

Drag the red dots to change the shape of the triangle.


Right-click here to download this page and the Java Class File.


This applet is dedicated to the memory of Dwight Paine who passed away on 18 May 2001.
Dr Paine wrote the famous Triangle Rhyme which I first read in the Sep 1983 issue of the Mathematics Magazine.

Basic properties of a triangle:
• The medians (lines joining each vertex to the midpoint of the opposite side) meet at the centroid G.
• The angle bisectors meet at the incentre I, the centre of the incircle.
• The external angle bisectors meet at the excentres IA, IB & IC, the centres of the excircles.
• The perpendicular bisectors meet at the circumcentre ), the centre of the circumcircle.
• The altitudes meet at the orthocentre H.
• The feet of the medians A', B' & C', the feet of the altitudes D, E & F, and the midpoints of HA, HB & HC all lie on the 9-point circle.
• The orthocentre H, the circumcentre O, the 9-point centre N and the centroid G all lie on the Euler line. N is the midpoint of OH and OG:GH = 1:2.

Other properties of a triangle:
• The centroid G is the centre of gravity of the triangle ABC, considering ABC as a sheet of uniform thickness.
• The centroid G divides each median in the ratio 1:2.
• The lines A'B', A'C' & B'C' divides triangle ABC into 4 congruent triangles, each similar to ABC.
• The medians divide triangle ABC into 6 small triangles of equal area.
• The inradius r of the incircle satisfies rs = area of ABC, where s = semiperimeter = (a + b + c)/2.
• The exradius r_a of the excircle touching side BC satisfies r_a(s - a) = area of ABC. Likewise for the other excircles.
• 1/r_a + 1/r_b + 1/r_c = 1/r.
• The incentre I is the orthocentre of the triangle formed by IA, IB, IC.
• The circumradius R of the circumcircle satisfies 2R = a / sin A = b / sin B = c / sin C.
• 4R = r_a + r_b + r_c - r.
• O is the orthocentre of the triangle A'B'C'.
• The pedal triangle DEF forms the shortest circuit joining the 3 sides a, b & c.
• The distance from the vertex A to the orthocentre H is given by AH = 2R cos A. Likewise for the other vertices.
• The 9-point circle touches the incircle and the 3 excircles.
• The radius of the 9-point circle is half the circumradius.