Geometric Bites

Consider a cyclic quadrilateral: all four of its vertices lie on a circle. Join the centre of the circle to each vertex. This creates four isosceles triangles, each made from two radii and one side of the quadrilateral. Label the angles of the quadrilateral at the circumference by w, x, y and z. In each isosceles triangle, the angles at the base are equal, so the angle at the centre is: π − 2w, π − 2x, π − 2y, π − 2z. These four central angles meet at a point, so together they make one full turn: (π − 2w) + (π − 2x) + (π − 2y) + (π − 2z) = 2π. Rearranging gives: −2w − 2x − 2y − 2z + 4π = 2π so 2w + 2x + 2y + 2z = 2π and therefore w + x + y + z = π. From this, the opposite-angle relations in the quadrilateral follow directly: w + z = π − (x + y), z + y = π − (w + x). So in any cyclic quadrilateral, each pair of opposite angles adds up to π radians .

The angle–addition identities for sine and cosine often appear as algebraic formulas, but they can also be understood by combining two right triangles in a simple geometric construction. The calculations for the side lengths follow directly from the definitions of sine and cosine. sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y − sin x sin y Start with a right triangle of angle y and hypotenuse 1. From basic trigonometry, its horizontal and vertical sides are: cos y and sin y. Next, attach a second right triangle with angle x . Its hypotenuse is the side of length cos y from the first triangle, so its adjacent and opposite sides become: adjacent = cos x · cos y opposite = sin x · cos y Likewise, if the first triangle's vertical side sin y is used as a hypotenuse in a similar way, it contributes: adjacent = cos x · sin y opposite = sin x · sin y When the horizontal components are combined, they give the expression for cos(x + y...