What do the Purser's theorem says?

Question

Mathworld's entry on Purser's Theorem says the following:

Let $t, u$, and $v$ be the lengths of the tangents to a circle $C$ from the vertices of a triangle with sides of lengths $a, b$, and $c$. Then the condition that $C$ is tangent to the circumcircle of the triangle is that $$\pm at\pm bu\pm cv=0.$$

Does this mean that there are $n_1,n_2,n_3\in\{0,1\}$ such that

$$(-1)^{n_1}at+(-1)^{n_2}bu+(-1)^{n_3}cv=0$$

or do we require that all possible eight combinations of $\pm$ gives the expression as zero?

The latter is possible iff $at, bu, cv$ all vanish. – anomaly Jul 04 '15 at 20:26 — anomaly, Jul 04 '15 at 20:26

justseen · Answer 1 · 2015-07-04T20:15:56.233

0

There is some discussion in http://artofproblemsolving.com/community/c6h67398 . So at least the necessity case is either $n_1=0,n_2=n_3=1$ or $n_1=1,n_2=n_3=0$.

edited Jul 04 '15 at 20:15

answered Jul 04 '15 at 16:42

justseen

1

What do the Purser's theorem says?

1 Answers1