★ AIMS Think+ · Q13

Given:

Use the consistent labelling D on BC, E on CA, F on AB.

To Find:

A, B and C under that labelling.

Property:

2D = B + C, 2E = C + A, 2F = A + B, coordinate by coordinate.

Solution:

Adding the last two equations and subtracting the first gives A = E + F − D = $\left(1,7\right)$.

Similarly B = F + D − E = $\left(-1,-1\right)$.

C = D + E − F = $\left(11,3\right)$.

Check BC midpoint $\left(5,1\right)$, CA midpoint $\left(6,5\right)$, AB midpoint $\left(0,3\right)$.

Results:

Vertices $\left(1,7\right)$, $\left(-1,-1\right)$, $\left(11,3\right)$. A different assignment of midpoint names permutes vertex labels.