2015-25.md

course	course_year	question_number	tags	title	year
Geometry	IB	25	IB 2015 Geometry	Paper 3, Section $I$, F	2015

State the sine rule for spherical triangles.

Let $\Delta$ be a spherical triangle with vertices $A, B$, and $C$, with angles $\alpha, \beta$ and $\gamma$ at the respective vertices. Let $a, b$, and $c$ be the lengths of the edges $B C, A C$ and $A B$ respectively. Show that $b=c$ if and only if $\beta=\gamma$. [You may use the cosine rule for spherical triangles.] Show that this holds if and only if there exists a reflection $M$ such that $M(A)=A, M(B)=C$ and $M(C)=B$.

Are there equilateral triangles on the sphere? Justify your answer.