From the fundamental property of the Cissoid of Diocles we can obtain by inversion an interesting theorem concerning the parabola. In the figure of the Cissoid given in Salmon’s H. P. C. Art. 214, A M₁ = M R, whence A M₁ = A R - A M; or A R = A M + A M₁. Inverting from the cusp and representing the inverse points by the same letters, we have for the parabola
This result is interpreted as follows:—draw the circle of curvature at the vertex of a parabola; this circle is tangent to the ordinate B D which is equal to the abscissa A D; draw a line through A cutting the circle in R, the ordinate B D in M, and the parabola in M₁; then
Draw the circle with centre at D and radius A D; any chord of the parabola through the vertex is cut harmonically by the parabola, the circle, and the double ordinate through D.