• $\mathrm{△}BPM\backslash triangle\; BPM$ ist gleichschenklig ($\overline{BM}=\overline{PM}=r\backslash overline\{BM\}\; =\; \backslash overline\{PM\}\; =\; r$)

1. $\mathrm{\sphericalangle }PBM=\phi \backslash sphericalangle\{PBM\}\; =\; \backslash varphi$ (Basiswinkel)

2. $\mathrm{\sphericalangle }BMP=180\mathrm{°}-2\phi \backslash sphericalangle\{BMP\}\; =\; 180°-\; 2\backslash varphi$ (Winkelsumme im Dreieck)

3. $\mu +\mathrm{\sphericalangle }BMP=180\mathrm{°}\backslash mu\; +\; \backslash sphericalangle\; \{BMP\}=\; 180°$ (Nebenwinkel)$\mu +\text{180°}-2\phi =180\mathrm{°}\backslash mu\; +\; \backslash text\{180°\}\; -\; 2\backslash varphi\; =\; 180°$

4. $\mu =2\phi \backslash mu\; =\; 2\backslash varphi$

5. $\mathrm{△}ABM\backslash triangle\{ABM\}$ ist gleichschenklig ($\overline{AM}=\overline{BM}=r\backslash overline\{AM\}\; =\; \backslash overline\{BM\}\; =\; r$) mit Basiswinkel $\alpha \backslash alpha$

6. Winkelsumme im Dreieck ABM:$\alpha +\alpha +2\phi =180\mathrm{°}\backslash alpha\; +\; \backslash alpha+\; 2\backslash varphi\; =180°$ $\mathrm{\mid }\backslash vert$:2

$\alpha +\phi =90\mathrm{°}\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ,\backslash alpha\; +\backslash varphi\; =\; 90°$ $\mathrm{\mid }\backslash vert$ $-\phi -\backslash varphi\%$

$\alpha =90\mathrm{°}-\phi \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;\backslash ,\backslash ;\backslash alpha=90°-\backslash varphi$

• $90\mathrm{°}+\alpha +\tau =180\mathrm{°}90°+\backslash alpha\; +\backslash tau\; =180°$ (gestreckter Winkel)$90\mathrm{°}+90\mathrm{°}-\phi +\tau =180\mathrm{°}90°+90°-\backslash varphi\; +\; \backslash tau\; =180°$

1. $\tau =\phi \backslash tau\; =\; \backslash varphi$