Tondeur’s text does not have many easy, practice problems, similar to what you can expect on quizzes and tests. So here is a growing collection of problems to test your understanding of the material.

You should also treat these problems as a practice quiz.

Question 1.

In the barycentric coordinates $(a,b,c)$ relative to $\triangle ABC$ where does the line $a = 0.3$ cross $(AB)$ ?

Substitute into $X=aA+bB+cC$ what is given. Then $X = 0.3A + bB + 0C$. Because $a+b+c=1$ you can solve for $b = 0.7$.

Question 2.

Let $A' = -B +2C, B'= 0.5 A + 0.5 C$ find $C'(AB)$ so that the three cevians are concurrent.

There is an easy ways of solving this problem. Using Ceva’s theorem, you can find the value of $\frac{C'-A}{C'-B}$ and solve that for the barycentric coordinates of $C'$. Be sure you know how to sketch the corresponding figure also freehand, not with KSEG.

A second solution is to express $(A A')$ $(B B')$ in parametric form (be sure to use different letters for the parameter.) Then find $G=(A A')(B B')$ by getting three equations in two unknowns (remember, barycentric coordinates are unique.) But they have a unique solution. Now repeat the process for $C' = (AB)(CG)$.

By doing this problem both ways, you are checking your arithmetic, which is easy to make mistakes in. Which method is less work?

Question 3.

Consider barycentric coordinates $(a,b,c)$ relative to $\triangle ABC$. Where does the line with equation $2b - c =0$ cross $(BC)$ ?

The point $X = aA + bB + cC$ must have $a = 0$ and $c=2b$. Substituting and remembering that $1 = a+b+c$ you can solve the three equations for a unique answer. Be sure you know how to sketch a believable figure for this. You can estimate the ratios by sight.