A Diagram for the Study and Solution of Triangles

Extract

The diagram which it is the purpose of these notes to explain is so simple in idea that it is surprising that it is not well known. Its idea is merely that the shape of a triangle depends only on two data and can therefore be represented in various ways by the coordinates of a point. To do this, suppose the scale of any triangle reduced so that its largest side is unity and let the other sides be x and y in descending order, then the point whose coordinates are x and y will represent the triangle. Moreover x and y are limited by the inequalities 1 ≥ x, x ≥ y, x + y ≥ 1, so the point representing any triangle lies in the quarter of the unit square shown in the diagram. Of this the boundary line x = 1 corresponds to isosceles triangles whose greater sides are equal, the line x = y to isosceles triangles whose smaller sides are equal, and the third line x + y = 1 to nugatory triangles with two angles equal to zero. The circle x² + y² = 1 corresponds to right-angled triangles, and triangles are obtuse-angled or acute-angled according as their representative points lie inside or outside this circle.