Circle inscribed in a triangle with triangle inscribed circle...

An equilateral triangle is inscribed in a circle, which (the circle) is also inscribed in another equilateral triangle. what is the ratio of the areas of the inner triangle and outside triangle?

Refer here for the figure. Let \(O\) be the center of the circle and the radius of the circle be \(r\). The equilateral triangle \(\Delta ABC\) is inscribed inside the circle and the circle is inscribed inside the equilateral triangle \(\Delta DEF\). Let \(AA'\) be perpendicular to \(BC\) and \(DD'\) be perpendicular to \(EF\). Since \(\Delta ABC\) and \(\Delta DEF\) are equilateral triangle, \(AA'\) and \(DD'\) are both medians to the respective sides from respective vertices and both pass through \(O\), which is also the centroid. Recall that the centroid divides the medians in the ratio \(2:1\). Hence, we get that
\[\dfrac{AO}{AA'} = \dfrac{DO}{DD'} = \dfrac23 \implies AA' = \dfrac{3r}2; DD' = 3r\]
Hence,
\[\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta DEF} = \dfrac{\dfrac12 \cdot AA' \cdot BC}{\dfrac12 \cdot DD' \cdot EF} = \dfrac{AA'}{DD'} \cdot \dfrac{BC}{EF} = \dfrac{AA'}{DD'} \cdot \dfrac{A'C}{D'F} = \left(\dfrac{AA'}{DD'} \right)^2 = \dfrac14\]

Don't you know that \(r=\frac{A}{s}\) if the circle is inscribed in a triangle. With r=radius of circle, A=area of triangle and s=a half of triangle's perimeter...
\(r=\frac{abc}{4A}\) if the circle is circumcircle with a,b,c=side of the triangle. Or I'll give the link for you? Sorry I can't draw the picture for you.

it can be done by rotation, rotate the inner triangle by 60 degrees with centroid fixed, so now the inner triangle has its vertices on the midpoints of the smaller triangle!! so 1/4 is tthe answer.

## Comments

Sort by:

TopNewestThe ratio is \(1/4\). Simply rotate the smaller triangle so that its corners lie at the midpoints of the sides of the larger triangle.

Log in to reply

i like yr answer

Log in to reply

Refer here for the figure. Let \(O\) be the center of the circle and the radius of the circle be \(r\). The equilateral triangle \(\Delta ABC\) is inscribed inside the circle and the circle is inscribed inside the equilateral triangle \(\Delta DEF\). Let \(AA'\) be perpendicular to \(BC\) and \(DD'\) be perpendicular to \(EF\). Since \(\Delta ABC\) and \(\Delta DEF\) are equilateral triangle, \(AA'\) and \(DD'\) are both medians to the respective sides from respective vertices and both pass through \(O\), which is also the centroid. Recall that the centroid divides the medians in the ratio \(2:1\). Hence, we get that \[\dfrac{AO}{AA'} = \dfrac{DO}{DD'} = \dfrac23 \implies AA' = \dfrac{3r}2; DD' = 3r\] Hence, \[\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta DEF} = \dfrac{\dfrac12 \cdot AA' \cdot BC}{\dfrac12 \cdot DD' \cdot EF} = \dfrac{AA'}{DD'} \cdot \dfrac{BC}{EF} = \dfrac{AA'}{DD'} \cdot \dfrac{A'C}{D'F} = \left(\dfrac{AA'}{DD'} \right)^2 = \dfrac14\]

Log in to reply

i think the ratio is 1/4

Log in to reply

I think it's \(\frac{1}{4}\)...

or maybe I have made wrong in calculations...

Log in to reply

Thanks, but could you give me steps on how you did it?

Log in to reply

Don't you know that \(r=\frac{A}{s}\) if the circle is inscribed in a triangle. With r=radius of circle, A=area of triangle and s=a half of triangle's perimeter... \(r=\frac{abc}{4A}\) if the circle is circumcircle with a,b,c=side of the triangle. Or I'll give the link for you? Sorry I can't draw the picture for you.

Log in to reply

according to me it should be 1/4

Log in to reply

\(\frac{1}{4}\)

Log in to reply

it can be done by rotation, rotate the inner triangle by 60 degrees with centroid fixed, so now the inner triangle has its vertices on the midpoints of the smaller triangle!! so 1/4 is tthe answer.

Log in to reply

1/4

Log in to reply

1/4

Log in to reply

It's 1/4.

Log in to reply

\frac{1}{4}

Log in to reply

Two triangles are similar, altitudes are 3R and 3R/2 => ratio of altitudes = 1 : 2

=> ratio of areas = 1 : 4

Log in to reply

Rotate the inner triangle so that it is the medial triangle of the outside triangle.

Log in to reply

Yeah.... I'll go with 1:4

Log in to reply

1/5

Log in to reply

1:3

Log in to reply

Fail.

Log in to reply