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.

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## Comments

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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.

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i like yr answer

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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, \[\dfrac{\text{Area of }\Delta ABC}{\text{Area of }\Delta DEF} = \dfrac{\dfrac12 \cdot AA' \cdot BC}{\dfrac12 \cdot DD' \cdot EF} = \dfrac{AA'}{DD'} \cdot \dfrac{BC}{EF} = \dfrac{AA'}{DD'} \cdot \dfrac{A'C}{D'F} = \left(\dfrac{AA'}{DD'} \right)^2 = \dfrac14\]

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

or maybe I have made wrong in calculations...

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

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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.

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i think the ratio is 1/4

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

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

=> ratio of areas = 1 : 4

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\frac{1}{4}

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It's 1/4.

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1/4

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1/4

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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.

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

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according to me it should be 1/4

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

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1/5

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1:3

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Fail.

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