Vectors and dot product

Multiply the vector lengths by the cosine of their angle, retaining the cosine's sign.

Rearrange the dot-product formula for the cosine. Divide the dot product by the product of both lengths and reduce the fraction.

Write the squared length as (u + v) · (u + v). Expand the dot product, then take the square root.

The signed projection is |u| cos α. Relate it to the dot product and divide by the length of the vector defining the projection direction.

Perpendicular vectors have zero dot product. Expand (u + tv) · v and remember that v · v is the squared length.

Expand both squared diagonal lengths using dot products. Subtract the equations so that the squared side lengths cancel.

Expand the squared length to obtain a quadratic in t with a positive leading coefficient. Complete the square or find the parabola's vertex.

The vertex is B, so use vectors BA and BC. The sign of their dot product determines whether the angle is less than, equal to or greater than 90°.

Distribute the dot product across all terms. Replace u · u and v · v by squared lengths and combine the mixed terms.

Let AB = u and AD = v. Express AM and AN using midpoint relations, then expand their dot product. Use the given angle's cosine for the mixed term.