Half ratio times length

The derivation of the formula

(1)   \begin{equation*} m_A=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC \end{equation*}

will be outlined in example 9, and will span (about) nine lines. Meanwhile, the formula and the title are not quite aligned \cdots

Copious verification

In equation (1) above, m_A is the length of the median from vertex A, and BC is the length of side BC. Our first four examples confirm that the relation (1) should hold for all values of r, except the unwanted ones: -1,0,1.

Example

Consider \triangle ABC with vertices at A\left(1,\frac{9}{4}\left), B\left(\frac{5}{2},\frac{15}{4}\right), and C(0,0). Find the length of the median from vertex A, and verify that equation (1) is satisfied.

The slopes of sides AB,BC,CA are 1,\frac{3}{2},\frac{9}{4}, respectively. They form a geometric progression in which r=\frac{3}{2}.

Let’s first find the length of the median from vertex A. The midpoint of side BC is at \left(\frac{5}{4},\frac{15}{8}\right). Together with A\left(1,\frac{9}{4}\left), the distance formula gives:

    \begin{equation*} \begin{split} m_A^2&=\left(1-\frac{5}{4}\right)^2+\left(\frac{9}{4}-\frac{15}{8}\right)^2\\ &=\left(-\frac{1}{4}\right)^2+\left(\frac{3}{8}\right)^2\\ &=\frac{13}{64}\\ \therefore m_A&=\frac{1}{8}\sqrt{13} \end{split} \end{equation*}

Thus, the length of the median from vertex A is m_A=\frac{1}{8}\sqrt{13}.

Next, the length of side BC is, by the distance formula:

    \begin{equation*} \begin{split} BC^2&=\left(\frac{5}{2}-0\right)^2+\left(\frac{15}{4}-0\right)^2\\ &=\frac{25}{4}+\frac{225}{16}\\ &=\frac{325}{16}\\ \therefore BC&=\frac{5}{4}\sqrt{13} \end{split} \end{equation*}

Finally, we check that both m_A=\frac{1}{8}\sqrt{13} and BC=\frac{5}{4}\sqrt{13} are related via equation (1):

    \begin{equation*} \begin{split} m_A&=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC\\ &=\frac{1}{2}\left(\frac{3/2-1}{3/2+1}\right)\frac{5}{4}\sqrt{13}\\ &=\frac{1}{2}\left(\frac{1/2}{5/2}\right)\frac{5}{4}\sqrt{13}\\ &=\frac{1}{8}\sqrt{13} \end{split} \end{equation*}

Expected.

Example

Consider \triangle ABC with vertices at A\left(-\frac{7}{4},-\frac{15}{2}\right), B\left(\frac{1}{2},-3\right), and C(-1,6). Find the length of the median from vertex A and verify that equation (1) is satisfied.

The slopes of sides AB,BC,CA are 2,-6,18, respectively. They form a geometric progression in which r=-3. Special, special case.

The midpoint of side BC is at \left(-\frac{1}{4},\frac{3}{2}\right). Together with A\left(-\frac{7}{4},-\frac{15}{2}\right), we can find the length of the median from vertex A:

    \begin{equation*} \begin{split} m_A^2&=\left(-\frac{7}{4}+\frac{1}{4}\right)^2+\left(-\frac{15}{2}-\frac{3}{2}\right)^2\\ &=\left(-\frac{3}{2}\right)^2+\left(-\frac{18}{2}\right)^2\\ &=\frac{333}{4}\\ \therefore m_A&=\frac{3}{2}\sqrt{37} \end{split} \end{equation*}

With B at \left(\frac{1}{2},-3\right) and C at (-1,6):

    \begin{equation*} \begin{split} BC^2&=\left(-1-\frac{1}{2}\right)^2+\left(6+3\right)^2\\ &=\frac{9}{4}+81\\ &=\frac{333}{4}\\ \therefore BC&=\frac{3}{2}\sqrt{37} \end{split} \end{equation*}

Surprised — that m_A=BC.

If we put r=-3 in equation (1) we see that

    \[m_A=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC=\frac{1}{2}\left(\frac{-3-1}{-3+1}\right)BC=\frac{1}{2}\left(\frac{-4}{-2}\right)BC=BC.\]

Surprise no longer applies. Subtract surprise from your dictionary.

If the slopes of the sides of a triangle form a geometric progression with common ration r=-3, then one median and the side opposite it are equal in length.

Example

Consider \triangle ABC with vertices at A(1,2\sqrt{2}), B(-1,\sqrt{2}), and C(0,0). Find the length of the median from vertex A, and verify that equation (1) is satisfied.

The slopes of sides AB,BC,CA are \frac{\sqrt{2}}{2},-\sqrt{2},2\sqrt{2}, respectively. They form a geometric progression in which r=-2.

The midpoint of side BC is at \left(-\frac{1}{2},\frac{\sqrt{2}}{2}\right). Together with A\left(1,2\sqrt{2}\right), we find:

    \begin{equation*} \begin{split} m_A^2&=\left(1+\frac{1}{2}\right)^2+\left(2\sqrt{2}-\frac{\sqrt{2}}{2}\right)^2\\ &=\frac{9}{4}+\frac{18}{4}\\ &=\frac{27}{4}\\ \therefore m_A&=\frac{3}{2}\sqrt{3} \end{split} \end{equation*}

With B at \left(-1,\sqrt{2}\right) and C at (0,0):

    \begin{equation*} \begin{split} BC^2&=\left(-1-0\right)^2+\left(\sqrt{2}-0\right)^2\\ &=3\\ \therefore BC&=\sqrt{3} \end{split} \end{equation*}

Put r=-2 in equation (1):

    \begin{equation*} \begin{split} m_A&=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC\\ &=\frac{1}{2}\left(\frac{-2-1}{-2+1}\right)\sqrt{3}\\ &=\frac{1}{2}\times 3\times \sqrt{3}\\ &=\frac{3}{2}\sqrt{3} \end{split} \end{equation*}

YESpected.

Example

Consider \triangle ABC with vertices at A(1,4), B(3,6), and C(0,0). Find the length of the median from vertex A, and verify that equation (1) is satisfied.

Our examples will be incomplete without this ugly, lovely triangle.

The midpoint of side BC is at \left(\frac{3}{2},3\right). Together with A(1,4):

    \begin{equation*} \begin{split} m_A^2&=\left(1-\frac{3}{2}\right)^2+\left(4-3\right)^2\\ &=\frac{1}{4}+1\\ &=\frac{5}{4}\\ \therefore m_A&=\frac{1}{2}\sqrt{5} \end{split} \end{equation*}

With B at (3,6) and C at (0,0), the length of BC is seen to be

    \[BC=\sqrt{3^2+6^2}=\sqrt{45}=3\sqrt{5}.\]

Since the slopes of the sides of \triangle ABC form a geometric progression with common ratio r=2, let’s put r=2 in equation (1):

    \begin{equation*} \begin{split} m_A&=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC\\ &=\frac{1}{2}\left(\frac{2-1}{2+1}\right)3\sqrt{5}\\ &=\frac{1}{2}\times\frac{1}{3}\times 3\sqrt{5}\\ &=\frac{1}{2}\sqrt{5} \end{split} \end{equation*}

Passed.

Obvious observations

Maybe you’ve wondered why we omitted absolute value in equation (1):

    \[m_A=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC.\]

Wonder no more.

Example

If r> 1, PROVE that \frac{r-1}{r+1}> 0.

This is because r> 1\implies r-1> 0, and r+1> 2> 0, so the quotient \frac{r-1}{r+1} is also greater than 0.

Example

If r< -1, PROVE that \frac{r-1}{r+1}> 0.

If r< -1, then we have r-1< -2< 0 and also r+1< 0. So the quotient \frac{r-1}{r+1}> 0.

Example

If 0< r< 1, PROVE that \frac{r-1}{r+1}< 0

If r< 1, then r-1< 0. The fact that 0< r ensures that r+1> 0. So the numerator and denominator of \frac{r-1}{r+1} have negative and positive signs, respectively. This makes the quotient negative.

However, if the common ratio of our three-term geometric progression happens to be sandwiched between 0 and 1, we can simply use its reciprocal — which will then satisfy \frac{1}{r}> 1 — in equation (1). This is the case of Example 5.

Example

If -1< r< 0, PROVE that \frac{r-1}{r+1}< 0.

If r< 0, then r-1< -1< 0. If in addition -1< r, then adding 1 to both sides of the inequality gives 0< r+1. So the numerator and denominator of \frac{r-1}{r+1} are negative and positive, respectively. This makes the quotient to be negative.

However, if the common ratio of our three-term geometric progression happens to be sandwiched between -1 and 0, then we can simply use its reciprocal — which will then satisfy \frac{1}{r}< -1 — in equation (1). This is the case of Example 6.

Tedious derivation

As we derive equation (1) in what follows, are we really being serious by hinting that the procedure is tedious? Follow us to see that the answer is obvious.

PROVE that equation (1) holds for any triangle ABC with slopes a,ar,ar^2 for the sides AB,BC,CA.

We’ll use two identities from our previous posts:

(2)   \begin{equation*} \begin{split} m_B^2+m_C^2&=\left(\frac{(r+2)^2+(2r+1)^2}{(r-1)^2}\right)m_A^2\\ AB^2+AC^2&=\frac{r^2+1}{(r+1)^2}BC^2\\ \end{split} \end{equation*}

and a well-known identity, namely

    \[m_a^2+m_b^2+m_c^2=\frac{3}{4}(a^2+b^2+c^2),\]

which holds for any triangle. We’ll write the latter in the form

(3)   \begin{equation*}m_A^2+m_B^2+m_C^2&=\frac{3}{4}(AB^2+BC^2+CA^2).\end{equation*}

Eliminate m_B,m_C,AB,AC from equations (2) and (3):

    \begin{equation*} \left(\frac{(r+2)^2+(2r+1)^2}{(r-1)^2}+1\right)m_A^2&=\frac{3}{4}\left(\frac{r^2+1}{(r+1)^2}+1\right)BC^2\\ \end{equation*}

Isolate m_A^2 and continue the algebra:

    \begin{equation*} \begin{split} m_A^2&=\frac{3}{4}\left(\frac{r^2+1+(r+1)^2}{(r+1)^2}\right)\times\left(\frac{(r-1)^2}{(r+2)^2+(2r+1)^2+(r-1)^2}\right)BC^2\\ &=\frac{3}{4}\left(\frac{2(r^2+r+1)}{(r+1)^2}\right)\times \left(\frac{(r-1)^2}{6(r^2+r+1)}\right)BC^2\\ &=\frac{1}{4}\left(\frac{r-1}{r+1}\right)^2BC^2\\ \therefore m_A&=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC \end{split} \end{equation*}

Just as we said.

Find coordinates for the vertices A,B,C of a \triangle ABC in which the length of one median is one-third of the length of the opposite side.

We want a triangle ABC in which m_A=\frac{1}{3}BC, for example. Without recourse to equation (1), solving this problem maybe somewhat random.

So we’ll use what we’ve got. In (1), set

    \[\frac{1}{2}\left(\frac{r-1}{r+1}\right)=\frac{1}{3}\]

and solve for r: \quad 3r-3=2r+2\implies r=5.

It remains to find coordinates for the vertices of a triangle in which the slopes of the sides form a geometric progression with common ratio r=5.

Use the most basic set of coordinates for the vertices:

    \begin{equation*} \begin{split} &(0,0)\\ &(1,r^2)\\ &(1+r,r+r^2)\\ &\vdots\vdots\vdots\\ &C(0,0)\\ &A(1,25)\\ &B(6,30) \end{split} \end{equation*}

That’s it. \triangle ABC with vertices above is what we want:

    \begin{equation*} \begin{split} BC^2&=6^2+30^2\\ \therefore BC&=6\sqrt{26}\\ &\vdots\vdots\\ m_A^2&=\left(1-3\right)^2+\left(25-15\right)^2\\ &=104\\ \therefore m_A&=2\sqrt{26}\\ &=\frac{1}{3}\times 6\sqrt{26}\\ &=\frac{1}{3}BC\\ \vdots\vdots\ddots&\ddots\vdots\vdots\\ \end{split} \end{equation*}

Takeaway

Our main equation (1)

    \[m_A=\frac{1}{2}\left(\frac{r-1}{r+1}\right)BC\]

is similar to what happens in a right triangle, where the length of the median to the hypotenuse is half of the length of the hypotenuse.

Tasks

  1. Find coordinates for the vertices of \triangle ABC in which one median has length equal to \frac{3}{5} of the length of the opposite side.
  2. Find coordinates for the vertices of a right triangle in which the side-slopes form a geometric progression with common ratio r=-3.
  3. (Polynomial equations) Let \triangle ABC be such that sides AB,BC,CA have slopes a,ar,ar^2, respectively. PROVE that:
    • a^2r^4+2a^2r^3+2r+1=0, if l_1=l_2;
    • a^2r^4-a^2r^2-r^2+1=0, if l_1=l_3;
    • 2a^2r^3+a^2r^2+r^2+2r=0, if l_2=l_3.
  4. (Confusing notation) In any \triangle ABC with vertices A(x_1,y_1), B(x_2,y_2), and C(x_3,y_3), let the slopes of the medians be denoted by m_A,m_B,m_C, and let the slopes of the sides be m_{AB},m_{BC},m_{CA}. PROVE that:
    • x_1(2m_A-m_B-m_C)+x_2(2m_B-m_A-m_C)+x_3(2m_C-m_A-m_B)=0;
    • x_1(m_{AB}-m_{CA})+x_2(m_{BC}-m_{AB})+x_3(m_{CA}-m_{BC})=0.
      (In our slipshod style, we sometimes use m_A to denote the length of a median and sometimes the slope of the same median. Are we short of notations?)
  5. (Alternative derivation) Let \triangle ABC be equilateral such that sides AB,BC,CA have slopes a,ar,ar^2, respectively. PROVE that:
    • (ar)^2=1;
    • r^2+4r+1=0.
      (You can apply exercise 3 above, which uses a different approach compared to what we did in a previous post.)
  6. (Linear combinations) Let A(x_1,y_1), B(x_2,y_2), and C(x_3,y_3) be the vertices of \triangle ABC in which sides AB,BC,CA have slopes a,-3a,9a. PROVE that 2x_1+x_2-3x_3=0, and 2y_1-3y_2+y_3=0.
  7. Let -1\neq r\in\mathbb{Z}. PROVE that \frac{1}{2}\left(\frac{r-1}{r+1}\right) is also an integer if, and only if, r=-3,1.
  8. Let 0,1,-1\neq r\in\mathbb{Z} and consider \triangle ABC with slopes a,ar,ar^2 for sides AB,BC,CA. PROVE that the length m_A of the median from vertex A is an integer multiple of the length of side BC if, and only if, r=-3.
  9. Find appropriate coordinates for the vertices of \triangle ABC in which the side-slopes form a geometric progression and the length of the median from vertex A is twice the length of side BC.
  10. Give an example of a triangle ABC in which the length of the median from vertex A is equal to the length of side BC, but the slopes of the sides do not form a geometric progression. (This shows that the converse of example 2 is not true.)