Plane Trigonometry

The Triangle and Circle

Let 𝑟 = radius of inscribed circle 𝑟_𝑎 = radius of eseribed circle touching the side 𝑎 𝑅 = radius of circumscribing circle 709 𝑟=△𝑠From Fig., △=𝑟𝑎2+𝑟𝑏2+𝑟𝑐2 710 𝑟=𝑎sin𝐵2sin𝐶2cos𝐴2By 𝑎=𝑟cot𝐵2+𝑟sin𝐶2 711 𝑟_𝑎=△𝑠−𝑎By △=𝑟_𝑎𝑏2+𝑟_𝑎𝑐2−𝑟_𝑎𝑎2 712 𝑟_𝑎=𝑎cos𝐵2cos𝐶2cos𝐴2By 𝑎=𝑟_𝑎tan𝐵2+𝑟tan𝐶2 713 𝑅=𝑎2sin𝐴=𝑎𝑏𝑐4△By III.20 and 706 715 𝑅=14(𝑏+𝑐)²sec²𝐴2+(𝑏−𝑐)²cosec²𝐴2702 716 Distance between the centres of inscribed and circumscribed circles. =𝑅²−2𝑅𝑟936 716 Radius of circle touching 𝑏, 𝑐 and the inscribed circle 𝑟′=𝑟tan²14(𝐵+𝐶)By sin𝐴2=𝑟−𝑟′𝑟+𝑟′

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive