PROP. IV.

The tangent at any point of a parabola bisects the angle between the focal distance of the point and the perpendicular from the point on the directrix.

Let PZ (fig. 7) be the tangent at P, meeting the directrix in Z ; then, if PM be drawn per pendicular to the directrix, it is easily seen that the two triangles SPZ, MPZ are equal in all respects, and the angle SPZ equal to the angle MPZ.

If SAI be joined, it can be shown that it is bisected at right angles by PZ, and that its middle point is the point Y in Prop. ii.

The line AY, it will be observed, is the tangent to the parabola at the vertex A.

It appears, therefore, that the locus of the foot of the perpendi, calar front the focus on the tangent at any point is the tangent at the vertex.

It can also be seen that, if the tangent at P meet the axis in '1', then SP= ST. For the angles STP, SPT are each equal to the angle MPT, and therefore (Encl. i, 6) SP, ST are equal.

It may further be remarked. that, if 0 be any point in the tangent at P, then the triangles SPO, 311'0 are equal in all respects.

If PN be drawn perpendicular to the axis to meet it in N, then it will be seen that PN = 2AY and TN - 2AN = 2AT.

Now, iu the right-angled triangle TYS, - TA ; AY = AY : AS (Encl. vi. 8), and therefore Y A.2 = TA , AS Therefore PN2 = 4YA2 = 4TA.. AS = 4AS . AN.

If the normal PG be drawn meeting the axis in G, then the triangles PNG, YAS are similar, and thereforeNG •AS = PN:YA 2:1 .•. NG - 2AS To draw a tangent to a parabola at a point on the curve.

First Method. - Take a point T in the axis (fig. 7), such that ST is equal to SP, and join TP. Then STP will be the tangent at P.

Second Method. - Draw SZ at right angles to SP, meeting the directrix in Z. ZP is the tangent at P.

Third Method. - On SP as diameter describe a circle ; this will touch the tangent at the vertex AY in a point Y. VP is the tangent rt P.