Prove concurrency of triangle altitudes with vector algebra?

Solution 1:

Denote the three vertices of the triangle by three column vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \ (\in\mathbb{R}^2)$ and the orthocenter by $\mathbf{x}$. By definition, $\mathbf{x}$ must satisfies $$ \begin{cases} (\mathbf{v}-\mathbf{w})\cdot(\mathbf{x}-\mathbf{u})=0\\ (\mathbf{w}-\mathbf{u})\cdot(\mathbf{x}-\mathbf{v})=0\\ (\mathbf{u}-\mathbf{v})\cdot(\mathbf{x}-\mathbf{w})=0 \end{cases} \ \Rightarrow \begin{cases} (\mathbf{v}-\mathbf{w})\cdot\mathbf{x}=(\mathbf{v}-\mathbf{w})\cdot\mathbf{u}\\ (\mathbf{w}-\mathbf{u})\cdot\mathbf{x}=(\mathbf{w}-\mathbf{u})\cdot\mathbf{v}\\ (\mathbf{u}-\mathbf{v})\cdot\mathbf{x}=(\mathbf{u}-\mathbf{v})\cdot\mathbf{w} \end{cases} $$ This can be rewritten in the matrix form of $A\mathbf{x}=\mathbf{b}$: $$ \underbrace{\begin{bmatrix}(\mathbf{v}-\mathbf{w})^T\\ (\mathbf{w}-\mathbf{u})^T\\ (\mathbf{u}-\mathbf{v})^T\end{bmatrix}}_{A} \ \mathbf{x} = \underbrace{\begin{bmatrix} (\mathbf{v}-\mathbf{w})\cdot\mathbf{u}\\ (\mathbf{w}-\mathbf{u})\cdot\mathbf{v}\\ (\mathbf{u}-\mathbf{v})\cdot\mathbf{w} \end{bmatrix}}_{\mathbf{b}} $$ So, the remaining question is, does there exist a unique solution for $A\mathbf{x}=\mathbf{b}$? Note that $A$ is a 3x2 matrix and $\mathbf{x}$ is a 2-vector, so the above system has three equations in two unknowns, i.e it is overdetermined. Can you show that each one of these three equations is a linear combination of the other two? Is it possible to remove one row from $A$, so that the remaining 2x2 submatrix is invertible?

Solution 2:

I am giving below a rather simple solution using vectors. Please make your own figure as I could not put in the same. Also treat small letters $a$, $b$, and $c$ as vectors. Write to me if you need more explanation.

Let $ABC$ a be triangle, let the perpendicular from $A$ to $BC$ meet $BC$ at $D$, and let the perpendicular from $B$ to $AC$ meet $AC$ at $E$. Let the lines $AD$ and $BE$ intersect at $O$. Let $O$ be the origin.

To prove: if we join $C$ and $O$ and extend it up to meet $AB$ at $F$, then $CF$ is perpendicular to $AB$.

Now, $CB$ is perpendicular to $OA$ (part of $AD$).

Therefore (1) $(c-b) \cdot a = 0$, since $(c-b)$ is the vector $CB$ and $OA$ is the vector $a$.

Similarly, $CA$ is perpendicular to $OB$ (part of $BE$).

Therefore, (2) $(c-a) \cdot b = 0$.

From (1), we obtain (3) $c \cdot a - b \cdot a = 0$.

From (2), we obtain (4) $c \cdot b - a \cdot b = 0$.

Subtracting (3) from (4), we get

$(c \cdot b - a \cdot b) – (c \cdot a - b \cdot a) = 0$

$\Rightarrow (c \cdot b - c \cdot a) – (a \cdot b - b \cdot a) = 0$

$\Rightarrow (c \cdot b - c \cdot a) = 0$, since $a \cdot b = b \cdot a$

$\Rightarrow (b - a) \cdot c = 0$.

Therefore $BA$ is perpendicular to $OC$ or $FC$. Therefore $O$ is the orthocenter of the triangle $ABC$ from which the three altitudes pass.

Solution 3:

Let's use the figure above in our approach.

Let be $|\vec{b}|=b$, $|\vec{c}|=c$, and $\vec{b}\cdot\vec{c}= m$. The orthocenter can be calculated in two ways: $$O=A+\overrightarrow{AB}+\lambda\overrightarrow{HB}\quad (1)$$ or $$O=A+\overrightarrow{AC}+\mu\overrightarrow{JC}\quad (2)$$ But $$\overrightarrow{HB}=\vec{c}-[\frac{(\vec{b}\cdot\vec{c})}{b}]\frac{\vec{b}}{b}\quad (3)$$ and $$\overrightarrow{JC}=\vec{b}-[\frac{(\vec{b}\cdot\vec{c})}{c}]\frac{\vec{c}}{c}\quad (4)$$ If we substitute $(3)$ and $(4)$ in $(1)$ and $(2)$, we get: $$O=A+\vec{c}+\lambda[\vec{c}-(\frac{m}{b^2})\vec{b}]\quad (5)$$ and $$O=A+\vec{b}+\mu[\vec{b}-(\frac{m}{c^2})\vec{c}]\quad (6)$$ Using equations $(5)$ and $(6)$ we get: $$\vec{c}+\lambda[\vec{c}-(\frac{m}{b^2})\vec{b}]=\vec{b}+\mu[\vec{b}-(\frac{m}{c^2})\vec{c}]\Rightarrow$$

$$\Rightarrow(1+\lambda)\vec{c}-\lambda(\frac{m}{b^2})\vec{b}=(1+\mu)\vec{b}-\mu(\frac{m}{c^2})\vec{c}\quad(7)$$

As $\vec{b}$ and $\vec{c}$ are linearly independent we can solve equation $(7)$ and we get: $$\mu=\frac{(m-b^2)}{c^2b^2-m^2}c^2$$

Now we can express $\overrightarrow{AO}$ as

$$\overrightarrow{AO}=\vec{b}+\frac{(m-b^2)}{c^2b^2-m^2}c^2[\vec{b}-(\frac{m}{c^2})\vec{c}]$$

If $\overrightarrow{AO}\cdot\overrightarrow{BC}=0$ then we can conclude that the three altitudes are concurrent.

So $$\overrightarrow{AO}\cdot\overrightarrow{BC}=(\vec{b}+\frac{(m-b^2)}{c^2b^2-m^2}c^2[\vec{b}-(\frac{m}{c^2})\vec{c}])\cdot(\vec{b}-\vec{c})\Rightarrow$$

$$\Rightarrow \overrightarrow{AO}\cdot\overrightarrow{BC}=b^2+\frac{(m-b^2)}{c^2b^2-m^2}c^2(b^2-\frac{m^2}{c^2})-m+\frac{(m-b^2)}{c^2b^2-m^2}c^2(-m+m)\Rightarrow$$

$$\Rightarrow \overrightarrow{AO}\cdot\overrightarrow{BC}=b^2+(m-b^2)-m+0\Rightarrow$$

$$\Rightarrow \overrightarrow{AO}\cdot\overrightarrow{BC}=0$$

Therefore the three altitudes are concurrent at point $O$.

Solution 4:

Assume that the altitudes $AI$ and $CJ$ intersect at point O. We need to prove that $BO$ is perpendicular to $AC$.

$\vec{BJ}$ is the projection of $\vec{BO}$ onto $\vec{BA}$. Hence:

$$ \vec{BJ} = \frac{\vec{BO}\cdot\vec{BA}}{||\vec{BA}||^2}\vec{BA} \quad (1) $$

However, $\vec{BJ}$ is also the projection of $\vec{BC}$ onto $\vec{BA}$. Hence:

$$ \vec{BJ} = \frac{\vec{BC}\cdot\vec{BA}}{||\vec{BA}||^2}\vec{BA} \quad (2) $$

From (1) and (2) we have that $\vec{BO}\cdot\vec{BA} = \vec{BC}\cdot\vec{BA}$. In the same way working with $\vec{BO}$ on $\vec{BC}$ we get $\vec{BO}\cdot\vec{BC} = \vec{BA}\cdot\vec{BC}$. Hence we have $\vec{BO}\cdot\vec{BA} = \vec{BC}\cdot\vec{BA} = \vec{BO}\cdot\vec{BC}$. Now:

$$ \vec{BO} \cdot \vec{AC} = \vec{BO} \cdot (\vec{AB} + \vec{BC}) = \vec{BO} \cdot \vec{BC} - \vec{BO} \cdot \vec{BA} = 0 $$

Hence $\vec{BO}$ and $\vec{AC}$ are perpendicular.