Substituting these quadrances into the equation to be proved:  Now, if and represent distinct points, such that , we may divide both sides by :  The lines (of quadrance ) and (of quadrance ) are perpendicular (their spread is 1) if and only if:  where is the quadrance between and . This is equivalent to the Pythagorean theorem (and its converse). There are many classical proofs of Pythagoras's theorem; this one is framed in the terms of rational trigonometry. The "spread" of an angle is the square of its sine. Given the triangle with a spread of 1 between sides and ,  where is the "quadrance", i.e. the square of the distance. Construct a line dividing the spread of 1, with the point on line , and making a spread of 1 with and . The triangles , and are similar (have the same spreads but not the same quadrances). This leads to two equations in ratios, based on the spreads of the sides of the triangle:  Now in general, the two spreads resulting from dividing a spread into two parts, as line does for spread , do not add up to the original spread since spread is a non-linear function. So we first prove that dividing a spread of 1, results in two spreads that do add up to the original spread of 1. For convenience, but with no loss of generality, we orient the lines intersecting with a spread of 1 to the coordinate axes, and label the dividing line with coordinates and . Then the two spreads are given by:  Hence  so that  Using the first two ratios from the first set of equations, this can be rewritten:  Multiplying both sides by :  Q.E.D. For any triangle with nonzero quadrances:  This is the law of sines, just squared. For any triangle ,  This is analogous to the law of cosines. It is called the 'cross law' because , the square of the cosine of the angle, is called the 'cross'. For any triangle ,  This relation can be derived from the formula for the sine of a compound angle: in a triangle (whose three angles sum to 180°) we have,  Equivalently, it describes the relationship between the spreads of three concurrent lines, as spread (like angle) is unaffected when the sides of a triangle are moved parallel to themselves to meet in a common point. Knowing two spreads allows the third to be calculated by solving the associated quadratic formula.