Triangle Solving – Problem 16: In , consider the statement: form an arithmetic progression and form a geometric progression

In $\triangle ABC$, consider the statement:

$A,B,C$ form an arithmetic progression and $\sin A,\sin B,\sin C$ form a geometric progression.

This statement is ( ) of: $\triangle ABC$ is equilateral.

A. Sufficient but not necessary B. Necessary but not sufficient C. Necessary and sufficient D. Neither sufficient nor necessary

Step 1. If $A,B,C$ form an arithmetic progression, then $2B=A+C$. Together with $A+B+C=\pi$, we get $B=\frac{\pi}{3}$.

Step 2. If $\sin A,\sin B,\sin C$ form a geometric progression, then $\sin^{2}B=\sin A\sin C$. By the Law of Sines this implies $b^{2}=ac$.

Step 3. By the Law of Cosines, $b^{2}=a^{2}+c^{2}-2ac\cos B$. With $B=\frac{\pi}{3}$, this becomes $b^{2}=a^{2}+c^{2}-ac$. Combining with $b^{2}=ac$ yields $a=c$, so $A=C$.

Step 4. Since $A+C=\pi-B=\frac{2\pi}{3}$ and $A=C$, we get $A=C=\frac{\pi}{3}$. Hence $A=B=C$ and the triangle is equilateral.

Step 5. Conversely, if the triangle is equilateral then $A=B=C=\frac{\pi}{3}$, so $A,B,C$ are in arithmetic progression and $\sin A=\sin B=\sin C$ are in geometric progression.

Step 6. Therefore the statement is necessary and sufficient, so the correct choice is C.

C