Proposition: 3.37: Converse of Tangent Secant Theorem

(Proposition 37 from Book 3 of Euclid's “Elements”)

If some point is taken outside a circle, and two straight lines radiate from the point towards the circle, and one of them cuts the circle, and the (other) meets (it), and the (rectangle contained) by the whole (straight line) cutting (the circle), and the (part of it) cut off outside (the circle), between the point and the convex circumference, is equal to the (square) on the (straight line) meeting (the circle), then the (straight line) meeting (the circle) will touch the circle.


Modern Formulation

If a straight line $DB$ intersects with a secant $DA$ of a given circle in $D$ and this secant cuts the circle at the points $C$ and $A$ such that for the lengths segments fulfill the inequality $|\overline{DC}|<|\overline{DA}|$, and if $|\overline{DC}|\cdot |\overline{DA}|=|\overline{DB}|^2,$ then $DB$ is a tangent $DB$ touching the circle at $D.$

Proofs: 1

Proofs: 1
Sections: 2

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016