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) $ ?

Answer. [show] ▶

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.

Answer. [show] ▶

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) $ ?

Answer. [show] ▶

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.