A circle tangent to a triangle side and to the two other extended sides. What is its diameter?

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## Solution

The diameter is 21.

This solution makes clever use of Heron’s formula to calculate the area of triangle ABC from its side lengths. It’s 84. Now note that this must be equal to the area of kites CHDE minus AIDE minus BHDI. This is readily calculated as (29+a+b)*r/2-ar-br. Since a+b=13, this becomes 8r. Thus 8r=84, and so 2r=21.

## Poem

The ice cream cone

A cone shaped pastry

Made of crispy wafer usually

A cone is baked flat and roled quickly

The cones can be brown,brittle and sugary

Crisp and flaky

You can get them in a creamery

Ice cream can be sweet, delicious and sweety

Children love it, it’s so hummy

## 3 replies on “The ice cream cone”

A little bit faster:

The area of ABC is equal to the area of

triangles ACD plus CBD minus ABD:

84 = (15*r + 14*r – 13*r)/2

84 = 8*r

21 = 2*r

Correct. Is much nicer!

21r=84+ra+rb

21r =84 + r (a+b)

21r = 84 +13r

8r= 84

2r= 21 ☺