Touching pentagons

Two regular pentagons are inscribed in a regular decagon in the way shown in the figure.
Did you know? The pentagons touch each other: points L and O are equal. Since version 5.0.393.0 GeoGebra is able to check this property symbolically as well. Just click on the Relation tool and select points L and O, finally choose "More...". (You may need to use Microsoft Edge to make this in your browser work.)


From L=O follows that these points are actually the center of the circumscribed circle of the decagon. Check this property by drawing some segments as the radius of the assumed circle and compare them by using the Relation tool. (This should work also in Google Chrome.)