Given a rhombus with side $a$ and acute angle $\alpha$.

Find the radius of the circle passing through two adjacent vertices of the rhombus and tangent to the opposite side of the rhombus or its extension.
Let $R$ be the radius of the circle passing through vertices $A$ and $B$ of the rhombus $ABCD (\angle A = \alpha)$ and tangent to the line $DC$ at point $M$. Let $K$ be the midpoint of $AB$, and $\angle MAB = \beta$. Then $KM = a \sin \alpha$, $AM^2 = AK^2 + KM^2 = a^2 / 4 + a^2 \sin 2 \alpha = 1 / 4 a^2 (1 + \sin^2 \alpha)$, $\sin \beta = KM / AM$, $R = AM / 2 \sin \beta = AM^2 / 2 KM = \frac{a (1 + 4 \sin^2 \alpha)}{8 \sin \alpha}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-39.jpg?height=434&width=540&top_left_y=0&top_left_x=765)

## Answer

$\frac{a (1 + 4 \sin^2 \alpha)}{8 \sin \alpha}$