Two points on sides AB and AC of a triangle

Question

How to determine using only the straightedge and compass the points P and Q on the sides AB and AC of a given triangle ABC such that the triangle APQ and the quadrilateral BPQC have the same surface and the segment PQ has the minimal possible length?

Answer 1

Draw a normal to $BC$ form $A$, call the length $\alpha$ . Draw another normal from $A$ to $PQ$, this will be called $\beta$ .

If $\frac{\alpha}{\beta} = s$, then $\frac{BC}{PQ} = s$ (Teaser : Find out the theorem yourself)

so, you want ($\frac{1}{2} \beta PQ = \frac{1}{2}(\alpha BC - \beta PQ)$).

Rewrite $BC$ and $\alpha$, and you get ($s^2 -1 = 1$) $ \implies s = \sqrt{2}$

Now you need to construct a segment $AP$ with $AP:PB = 1:\sqrt{2}-1$, and same with $q$.

Read this : http://www.mymathforum.com/viewtopic.php?f=13&t=32876 and this : http://www.mathopenref.com/constdividesegment.html

Answer 2

The least $PQ$ occurs when $AP = AQ$.

Let $AB=c$, $AC=b$ and $m(\angle A)=\alpha$.

We know that $Area_{APQ} = \frac{1}{2} Area_{ABC}$, then we have: $$\frac{1}{2} AP^2\sin\alpha =\frac{1}{2} \frac{1}{2} cb\sin\alpha \Rightarrow$$ $$AP=\sqrt{\frac{cb}{2}}.$$ Therefore $AP$ is the geometric mean of $c$ and $\frac{b}{2}$. We only have a solution if $\frac{c}{2} < b <2c$.

The construction is shown below:

EDIT.

(A short proof of the first statement)

Let $Area_{ABC}=S$, and $AP =x$ (where $S$ and $x$ are positive real numbers), then:

$$\frac{1}{2} AP \cdot AQ \sin\alpha = \frac{S}{2} \Rightarrow$$ $$\Rightarrow x AQ= \frac{S}{\sin\alpha} \Rightarrow$$ $$\Rightarrow AQ= \frac{S}{x \sin\alpha}. \quad (1)$$ But we know that $$PQ^2=AP^2 + AQ^2 -2AP \cdot PQ \cos \alpha \quad(2) $$ Substituting $(1)$ in $(2)$ we get: $$PQ^2= x^2 + (\frac{S}{x \sin\alpha})^2 - \frac{2S}{\tan \alpha} \Rightarrow $$ $$PQ= \sqrt{x^2 + (\frac{S}{x \sin\alpha})^2 - \frac{2S}{\tan \alpha}}. \quad(3) $$ At the minimal point the derivative of $PQ$ equals zero, so: $$2x+\frac{-2S^2}{x^3 \sin^2\alpha }=0 \Rightarrow$$ $$\Rightarrow x^4= \frac{S^2}{\sin^2\alpha} \Rightarrow$$ $$\Rightarrow x=\sqrt{\frac{S}{\sin\alpha}}. \quad(4)$$ Substituting $(4)$ in $(1)$, we get: $$AP=AQ=x=\sqrt{\frac{S}{\sin\alpha}}$$ Therefore the least $PQ$ occurs when $AP=AQ$.