Eritrea's Theorem

Question

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following:

Say you have a triangle, with sides of length $a$, $b$, and $c$. Draw the medians (lines $\overline{AG}$, $\overline{BI}$, and $\overline{CH}$ in the diagram), and the altitudes (lines $\overline{AD}$, $\overline{BF}$, and $\overline{CE}$ in the digram).

Call the distance between where the median and altitude hit a given side the sma of that side. In the diagram, the smas are $\overline{GD}$, $\overline{IF}$, and $\overline{HE}$.

Call the length of the sma on side $a$, $\alpha$. Similarly, on sides $b$ and $c$ we have smas $\beta$ and $\gamma$.

The theorem is: $$a\alpha+b\beta=c\gamma$$

In the picture, we have $5.19\times0.09+4.28\times0.9=4.39\times0.98$, which is true up to rounding error.

How would you prove this? I have almost no experience in geometry, so I wouldn't even know where to start on this. Thanks!

Answer 1

This is a different way to obtain the key equality $2a\alpha = c^2-b^2$.

We will use notion of power of a point with respect to a circle.

On the one hand, power of $G$ with respect to a circle centered at $I$ and radius $\frac b2$ equals $GI^2-\left(\frac b2\right)^2=\frac{c^2-b^2}{4}$.

On the other hand, this circle passes through $C$ and $D$ so the power of $G$ is $GD \cdot GC = \alpha \cdot \frac a2$ (assuming segments are oriented).

Thus $\frac{c^2-b^2}{4}=\alpha \cdot \frac a2$, so $2a \alpha = c^2-b^2$.

Answer 2

Express α, β, γ in terms of a, b, c

According to the Encyclopedia of Triangle Centers, the orthocenter $X(4)$ has barycentric coordinates $[\tan A:\tan B:\tan C]$. From the cosine law you have $\cos C=\frac{a^2+b^2-c^2}{2ab}$ and likewise for the other angles. So you get

\begin{align*} \tan C&=\frac{\sin C}{\cos C}=\frac{\sqrt{1-\cos C^2}}{\cos C} =\frac{\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}}{a^2+b^2-c^2} \\&=\frac{\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}}{a^2+b^2-c^2} \\&=\frac{4V}{a^2+b^2-c^2} \end{align*}

where $V$ denotes the area of the triangle, as obtained from Heron's formula. By canceling the $4V$ term and multiplying with all the denominators (which is allowed for homogeneous coordinates), you might write the barycentric coordinates of $X(4)$ also as

$$\begin{bmatrix} (a^2-b^2+c^2)(a^2+b^2-c^2)\\ (b^2-c^2+a^2)(b^2+c^2-a^2)\\ (c^2-a^2+b^2)(c^2+a^2-b^2) \end{bmatrix}$$

You get the footpoints of the heights by setting one of these coordinates to zero. So for example $D$ has barycentric coordinates

$$\begin{bmatrix} 0\\ (b^2-c^2+a^2)(b^2+c^2-a^2)\\ (c^2-a^2+b^2)(c^2+a^2-b^2) \end{bmatrix}\sim\begin{bmatrix} 0\\ b^2-c^2+a^2\\ c^2+a^2-b^2 \end{bmatrix}$$

which means you can write its Euclidean coordinates as

\begin{align*} D&=\frac{b^2-c^2+a^2}{(b^2-c^2+a^2)+(c^2+a^2-b^2)}B +\frac{c^2+a^2-b^2}{(b^2-c^2+a^2)+(c^2+a^2-b^2)}C \\&=\frac{a^2+b^2-c^2}{2a^2}B + \frac{a^2-b^2+c^2}{2a^2}C \\&=B + \frac{a^2-b^2+c^2}{2a^2}(C-B) \\&=B + \left(\frac12 + \frac{-b^2+c^2}{2a^2}\right)(C-B) \\&=G + \frac{-b^2+c^2}{2a^2}(C-B) \end{align*}

The distance between $D$ and $G$ is

$$\alpha=\lVert D-G\rVert = \left\lvert\frac{-b^2+c^2}{2a^2}\right\rvert\cdot\lVert C-B\rVert = \frac{\lvert b^2-c^2\rvert}{2a}\\ 2a\alpha=\lvert b^2-c^2\rvert$$

Orientation and order

Up to here, the above is an alternative to the shorter deduction timon92 posted in his answer. The discussion which follows below applies no matter how one obtains that formula for $2a\alpha$.

Using this formula, the equation of the theorem (multiplied by $2$ to simplify things) would be

$$\lvert b^2-c^2\rvert + \lvert c^2-a^2\rvert = \lvert a^2-b^2\rvert$$

This is not always the case. But if you use signed distances, e.g. always measured in counter-clockwise direction, you can omit the absolute values. Then write the formula as “sum equals zero” as columbus8myhw suggests in his comment, and you obtain

$$(b^2-c^2) + (c^2-a^2) + (a^2-b^2) = 0$$

which is obviously true.

If you prefer unsigned distances, when does the equation with those hold? It holds if and only if the difference inside the absolute value function has equal sign for both terms on the left hand side of the equation but opposite sign on the right. So you have two cases to consider:

\begin{gather*} b^2-c^2\ge0,\quad c^2-a^2\ge0,\quad a^2-b^2\le0 \quad\implies\quad b\ge c\ge a \\ b^2-c^2\le0,\quad c^2-a^2\le0,\quad a^2-b^2\ge0 \quad\implies\quad a\ge c\ge b \end{gather*}

That's what the “middle side” g.kov quoted in his comment refers to: the $c$ on the right hand side of the equation must be the side of median length.

Answer 3

After having reviewed the paper thoroughly, I revealed some important concept from the basic equation of the theorem. I am still working on the paper. $$\frac1{\tan\beta} = \frac1{\tan\alpha}+\frac1{\tan\theta},$$ where the angles $\beta, \alpha$ and $\theta$ are the angles that are formed respectively when the medians drawn to the middle side, the smallest side and the longest side are constructed. Therefore, this above stated equation is very helpful to draw the three medians. The theorem is very easy, basic and important as well.

Answer 4

I propose a generalization of the Eritrea theorem:

Answer 5

My friend let me that, he see the theorem appeared early from a book in Viet Nam, Author of the book is Nguyen Minh Ha, Publish since 2000. See vidu 5.22