Thread: Circumscribed isosceles triangle

An isosceles triangle with base 18 and legs 15 sits in a circle so that each of the edges touches the circumference. Find the radius of the circle.

I think this has something to do with similar triangles (constructing another flipped triangle under the base of the isosceles one), but I can't find the reason why the triangles would be similar. Looks like it has something to do with cyclic quads. Please help, thanks.

Originally Posted by DivideBy0

An isosceles triangle with base 18 and legs 15 sits in a circle so that each of the edges touches the circumference. Find the radius of the circle.

I think this has something to do with similar triangles (constructing another flipped triangle under the base of the isosceles one), but I can't find the reason why the triangles would be similar. Looks like it has something to do with cyclic quads. Please help, thanks.

Split isosceles down the middle into two right angled triangles.

Each with base 9 and hypotenuse 15

Angle opposite base is arcsin (9/15) = arcsin (3/5)

So angle at top of isosceles is 2 arcsin (3/5)

Join the centre of the circle to the two ends of the base forming a second isosceles triangle

By circle theorems angle at top of second isosceles triangle is twice that at top of first isosceles triangle (4 arcsin (3/5))

Split second isosceles triangle into two right angled triangles as we did with the first one

Each with base 9, no other side known and angle at top half of 4 arcsin (3/5)

ie 2 arcsin (3/5), call this A and radius r

Then sin A = r/9

r = 9 sin A = 9 sin (2 arcsin (3/5))