Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, concurrent$^\text {any}$ cevians in general) of a triangle are unique for a particular triangle.
I took a wild guess and I assumed Yes! Now assuming my guess is correct, I have the following questions
How can I construct a triangle given all three altitudes, medians or/and any three concurrent cevians, if it is possible?
N.B: If one looks closely at the order in which I wrote the question(the cevians coming last), the altitudes and the medians are special cases of concurrent cevians with properties
- The altitudes form an angle of $90°$ with the sides each of them touch.
- The medians bisect the side each of them touch.
With these properties it will be easier to construct the equivalent triangles (which I still don't know how) but with just any concurrent cevians, what other unique property can be added to make the construction possible? For example the angle they make with the sides they touch ($90°$ in the case of altitudes) or the ratio in which they divide the sides they touch ($1:1$ in the case of medians) or any other property for that matter.
EDIT
What André has shown below is a perfect example of three concurrent cevians forming two different triangles, thus given the lengths of three concurrent cevians, these cevians don't necessarily define a unique triangle. But also note that the altitudes of the equilateral triangle he defined are perpendicular to opposite sides while for the isosceles triangle, the altitude is, obviously also perpendicular to the opposite side with the remaining two cevians form approximately an angle of $50°$ with each opposite sides. Also note that the altitudes of the equilateral triangle bisects the opposite sides and the altitude of the isosceles triangle bisects its opposite sides while the remaining two cevians divides the opposite sides, each in the ratio $1:8$.
Now, given these additional properties (like the ratio of "bisection" or the angle formed with opposite sides) of these cevians, do they form a unique triangle (I'm assuming yes on this one) and if yes, how can one construct that unique triangle with a pair of compasses, a ruler and a protractor?
Best Answer
It is clear that the lengths of concurrent cevians cannot always determine the triangle. Indeed, they probably never can. But if it is clear, we must be able to give an explicit example.
Cevians $1$: Draw an equilateral triangle with height $1$. Pick as your cevians the altitudes.
Cevians $2$: Draw an isosceles triangle $ABC$ such that $AB=AC$, and $BC=\dfrac{10}{12}$, and the height of the triangle with respect to $A$ is equal to $1$. Then $AB=AC=\sqrt{1+(5/12)^2}=\dfrac{13}{12}$.
There are (unique) points $X$ and $Y$ on $AB$ and $AC$ respectively such that $BY=CX=1$. This is because as a point $T$ travels from $B$ to $A$ along $BA$, the length of $CT$ increases steadily from $\dfrac{10}{12}$ to $\dfrac{13}{12}$, so must be equal to $1$ somewhere between $B$ and $A$. Let $X$ be this value of $T$.
Let one of our cevians be the altitude from $A$, and let $BY$ and $CX$ be the other two cevians. These three cevians are concurrent because of the symmetry about the altitude from $A$, and they all have length $1$.