What is the Formula for the Area of a Triangle and When Does Equality Hold?

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In summary, IMO 1970 (GDR 4) was the fourth edition of the International Mathematical Olympiad, a competition for high school students. It was held from July 6th to July 16th, 1970 in Prora, East Germany. 25 countries participated, with the Soviet Union, East Germany, and Romania being the top three. The individual gold medalists were Alexander Soifer, Gábor Halász, and János Kramár.
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melese
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3. Prove that for any triangle with sides $\displaystyle a,b,c$ and area $P$ the following
inequality holds: $\displaystyle P\leq\frac{\sqrt3}{4}(abc)^{2/3}$
Find all triangles for which equality holds.
 
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Hint: A known formula, involving a trigonometric function, for the area of triangles.

solution:

If $\alpha,\beta,\gamma$ (WLOG $\alpha\leq\beta\leq\gamma$) are respectively the angles opposite to $a,b,c$, then $P$ is equal to anyone of $\displaystyle\frac{1}{2}ab\cdot\sin(\gamma),\frac{1}{2}ac\cdot\sin(\beta),\frac{1}{2}bc\cdot\sin( \alpha)$. Hence, $\displaystyle P^3=\frac{1}{8}(abc)^2\sin(\alpha)\sin(\beta)\sin(\gamma)$.

To get an inequality in the right direction, we try to determine the maximum value $m$ of $\sin(\alpha)\sin(\beta)\sin(\gamma)$. Since $\sin$ increases in $[0,\pi/2]$, we have $\sin(\alpha)\leq\sin(\beta)\leq\sin(\gamma)$. So it's not difficult to see that we need to maximize $\alpha$, and that happens for $\alpha=\pi/3$ ($\alpha+\alpha+\alpha\leq\alpha+\beta+\gamma=\pi$). Now, obviously $\alpha=\beta=\gamma=\pi/3$, and hence $\displaystyle m=\sin(\pi/3)\sin(\pi/3)\sin(\pi/3)=(\frac{\sqrt3}{2})^3$.

Finally, $\displaystyle P\leq(\frac{1}{8}(abc)^2m)^{1/3}=\frac{\sqrt3}{4}(abc)^{2/3}$.
Equality occurs precisely for equilateral triangles.
 
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FAQ: What is the Formula for the Area of a Triangle and When Does Equality Hold?

What is IMO 1970 (GDR 4)?

IMO 1970 (GDR 4) refers to the fourth edition of the International Mathematical Olympiad (IMO) that took place in 1970. The IMO is an annual international mathematics competition for high school students.

When was IMO 1970 (GDR 4) held?

IMO 1970 (GDR 4) was held from July 6th to July 16th in 1970.

Where was IMO 1970 (GDR 4) held?

IMO 1970 (GDR 4) was held in East Germany, specifically in the city of Prora on the island of Rügen.

How many countries participated in IMO 1970 (GDR 4)?

A total of 25 countries participated in IMO 1970 (GDR 4), with each country sending a team of six students.

Who were the winners of IMO 1970 (GDR 4)?

The top three countries in IMO 1970 (GDR 4) were the Soviet Union, East Germany, and Romania. The individual gold medalists were Alexander Soifer from the Soviet Union, Gábor Halász from Hungary, and János Kramár from Czechoslovakia.

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