Inscribed and Circumscribed Circles of a Triangle. Geometry video.

#geometry #circle #radius
Find the radius of the circumscribed circle of a triangle. Step by step instruction.

[Grizzly]

By the 'two tangents to a circle' theorem, △ABC must be equilateral.
By Vivani's Theorem, the height h of △ABC = 5 + 5 + 5 = 15
h = r₂ + r₁
∴ r₂ = h – r₁ = 15 – 5 = 10 units.

[easy_s]

In triangle BOD: BO²=BD²+OD²=BD²+25 so BD²=BO²-25.
Extend DO to point A. In triangle ABD: AB²=AD²+BD². With AB=2BD and AD=r₂+5 that gives (2BD)²=(r₂+5 )²+BD² which gives 4BD²=(r₂+5 )²+BD² and 3BD²=(r₂+5 )².
Now substitute for BD²=BO²-25 to get 3(BO²-25)=(r₂+5 )² which gives 3BO²-75=(r₂+5 )². BO=r₂ so 3r₂²-75=r₂²+10r₂+25 which simplifies to 2r₂²-10r₂-100=0 and r₂²-5r₂-50=0. Solve for r₂: (r₂+5)(r₂-10)=0 and so r₂=-5 or r₂=10. r₂=-5 is not a solution since r₂>0 so r₂=10.

[alfonsojesusdiaz]

Should you be so kind to explain the reason why
OB = 2OD ? .Bundles of thanks in advance.