Commentary on Proposition III.17¶
Back to Proposition III.17.
The construction shows, of course, that two straight lines can be drawn from a given external point to touch a given circle; and it is equally obvious that these two straight lines are equal in length and equally inclined to the straight line joining the external point to the centre of the given circle. These facts are given by Heron (an-Nairīzī, p. 130).
It is true that Euclid leaves out the case where the given point lies on the circumference of the circle, doubtless because the construction is so directly indicated by III. 16, Por. as to be scarcely worth a separate statement.
An easier solution is of course possible as soon as we know (III. 31) that the angle in a semicircle is a right angle; for we have only to describe a circle on AE as diameter, and this circle cuts the given circle in the two points of contact.