This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partialderivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in asystematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementingor decrementing its associated parameters by one. We apply the results to efficiently calculating the Laplacian ofpolynomial approximations of functions on the triangle, using polynomial degrees in the thousands, i.e., millions ofdegrees of freedom.