Hansen's problem

Solution method overview

Define the following angles: \begin{alignat}{5} \gamma &= \angle P_1 AP_2, &\quad \delta &= \angle P_1BP_2, \[4pt] \phi &= \angle P_2 AB, &\quad \psi &= \angle P_1 BA. \end{alignat} As a first step we will solve for φ and ψ. The sum of these two unknown angles is equal to the sum of *β*1 and *β*2, yielding the equation

\phi + \psi = \beta_1 + \beta_2.

A second equation can be found more laboriously, as follows. The law of sines yields

\frac{\overline{AB}}{\overline{P_2 B}} = \frac{\sin \alpha_2}{\sin \phi}, \qquad \frac{\overline{P_2 B}}{\overline{P_1 P_2}} = \frac{\sin \beta_1}{\sin \delta}.

Combining these, we get

\frac{\overline{AB}}{\overline{P_1 P_2}} = \frac{\sin \alpha_2 \sin \beta_1}{\sin \phi \sin \delta}.

Entirely analogous reasoning on the other side yields

\frac{\overline{AB}}{\overline{P_1 P_2}} = \frac{\sin \alpha_1 \sin \beta_2}{\sin \psi \sin \gamma}.

Setting these two equal gives

\frac{\sin \phi}{\sin \psi} = \frac{\sin \gamma \sin \alpha_2 \sin \beta_1}{\sin \delta \sin \alpha_1 \sin \beta_2} = k.

Using a known trigonometric identity this ratio of sines can be expressed as the tangent of an angle difference:

:\tan \tfrac12(\phi - \psi) = \frac{k-1}{k+1} \tan\tfrac12(\phi + \psi).

Where k = \frac{\sin \phi}{\sin \psi}.

This is the second equation we need. Once we solve the two equations for the two unknowns φ, ψ, we can use either of the two expressions above for \tfrac{\overline{AB}}{\overline{P_1 P_2}} to find since is known. We can then find all the other segments using the law of sines.

Solution algorithm

We are given four angles (α1, β1, α2, β2) and the distance . The calculation proceeds as follows:

• Calculate \begin{align} \gamma &= \pi-\alpha_1-\beta_1-\beta_2, \ \delta &= \pi-\alpha_2-\beta_1-\beta_2. \end{align}
• Calculate k = \frac{\sin \gamma \sin \alpha_2 \sin \beta_1}{\sin \delta \sin \alpha_1 \sin \beta_2}.
• Let s = \beta_1+\beta_2, \quad d = 2 \arctan \left( \frac{k-1}{k+1} \tan\tfrac12 s \right) and then \phi = \frac{s+d}{2}, \quad \psi = \frac{s-d}{2}.

Calculate \overline{P_1 P_2} = \overline{AB} \ \frac{\sin \phi \sin \delta}{\sin \alpha_2 \sin \beta_1} or equivalently \overline{P_1 P_2} = \overline{AB} \ \frac{\sin \psi \sin \gamma}{\sin \alpha_1 \sin \beta_2}. If one of these fractions has a denominator close to zero, use the other one.

References

References

1. Udo Hebisch: Ebene und Sphaerische Trigonometrie, Kapitel 1, Beispiel 4 (2005, 2006)[http://www.mathe.tu-freiberg.de/~hebisch/skripte/sphaerik/strigo.pdf] {{Webarchive. link. (2016-02-22)